Answer:
Step-by-step explanation:
When we divide two exponential expressions with the same base, we can subtract their exponents. Using this property, we can simplify the given expression as:
2^(x+2) /2^(x-1) = 2^x * 2^2 / 2^x * 2^(-1)
= 2^(x-1+2)
= 2^(x+1)
Therefore, the simplified expression is 2^(x+1).
Grace wants to buy a jump rope that costs $7, a board game that costs $10, and a playground ball that costs $4. She has saved $10 from her allowance, and her uncle gave her $3. How much more money does Grace need to buy the jump rope, the game, and the ball?
Grace need to buy the jump rope, the game, and the ball $8.
$7 will get you a jump rope.
$10 will get you a board game.
$4 will get you a playground ball.
total amount to be spent: $7 + $10 + $4 = $21
She has ten dollars.
$3 was all her uncle gave her.
13 dollars are all she has.
She needed $8, therefore 21 - 13 = $8.
a sum of money awarded as compensation, a bounty, or to cover costs. a wage that comes with a cost-of-living supplement. especially: a regular amount set aside for household or personal costs.
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PLEASE HELP MARKING BRAINLEIST JUST ANSWER ASAP
Answer:
The pink triangle
Step-by-step explanation:
IMPORTANT NOTE: Make sure all the units are the same and consistent
Perimeter of a figure = Total length of the outer boundary
Shape of each figure in this question = Isosceles Triangle
Perimeter of triangle = Sum of all three sides
Perimeter of pink triangle = 36m + 36m + 20m
= 92m
Perimeter of green triangle = 25m + 25m + 35m
= 85m
∴Comparing the two values calculated above, it can be observed that the pink triangle has a greater perimeter
indicate the method you would use to prove the two triangles ~=. if no method applies, enter none
The method we would be using here to prove the triangle's congruency is LA, the leg-acute theorem.
Define congruency?The dimensions of the sides and angles of two or more triangles determine whether they are congruent or not. A triangle's size and shape are determined by its three sides and three angles, respectively. If pairings of corresponding sides and corresponding angles are equal, two triangles are said to be congruent. They are the exact same size and form.
Here in the given figures, we can use the LA theorem also known as leg acute. It says that a right triangle is congruent if its leg and any acute angle are congruent with the corresponding leg and any acute angle of another right triangle.
The hypotenuse of the first triangle is corresponding to the hypotenuse of the second triangle.
The angle opposite to the right angle in both the triangles are corresponding to each other.
So as per the LA theorem we can prove that the triangle is congruent.
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what is the area of the parallelogram
The area of the parallelogram is 324 square yards.
What is a parallelogram ?
A parallelogram is a four-sided geometric shape that has two pairs of parallel sides. It isdefined by its four vertices, four sides, and two diagonals that intersect at their midpoint. The opposite sides of a parallelogram are congruent, and the opposite angles are also congruent. The adjacent angles are supplementary and add up to 180 degrees.
To find the area of a parallelogram, we need to multiply the length of the base by the height, which is the perpendicular distance from the base to the opposite side. This formula is similar to finding the area of a rectangle, where the base is one side of the rectangle and the height is the distance from that side to the opposite side.
Calculating the area of the given parallelogram :
The area of a parallelogram is A = bh, where b is the base and h is the height.
Given the height of the parallelogram is 12 yards and the base is 27 yards. Using the formula for the area of a parallelogram, we can calculate the area as follows:
A = bh
A = 27 yards × 12 yards
A = 324 square yards
Therefore, the area of the parallelogram is 324 square yards.
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Help!
I need your help.
Find the missing side of the triangle below
The value of y in the given triangle is 7.44 units.
What is tangent in trigonometry?The trigonometric ratio between the opposing and adjacent sides of a right triangle that contains an angle is its tangent. The trigonometric functions, also known as circular, angle, or goniometric functions in mathematics, are real functions that link the angle of a right-angled triangle to the ratios of its two side lengths. All geosciences, including navigation, solid mechanics, celestial mechanics, geodesy, and many more, utilise them extensively.
The given triangle is a right-triangle.
The trigonometric identity that gives the relationship between opposite side and adjacent side is tan.
Thus,
tan (28) = opposite / adjacent = y / 14
y = 0.53 (14)
y = 7.44
Hence, the value of y in the given triangle is 7.44 units.
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A right triangle is describe as having an angle of measure six less than negative two times a number, another angle measure that is three less than negative one-fourth the number, and a right angle. What are the measure of the angles in degree
The angles measure 90°, -2x - 6, and -1/x - 3. ⇒ x = -44. Therefore, the required measures of the angles are 90°, 82°, and 8° in the given triangle.
A right triangle is a type of triangle where one of the angles measures exactly 90 degrees. This angle is known as the right angle, and it is formed by the intersection of the two sides of the triangle that are perpendicular to each other. The other two angles of the right triangle are acute angles, meaning they measure less than 90 degrees.
The side opposite the right angle is called the hypotenuse, and it is always the longest side of the right triangle. The other two sides are called legs, and they can be of different lengths. This theorem is one of the most important and useful tools in geometry, and it allows us to solve many practical problems involving right triangles, such as finding the height of a building.
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a pastry chef accidentally inoculated a cream pie with six s. aureus cells. if s. aureus has a generation time of 60 minutes, how many cells would be in the cream pie after 7 hours?
After the time of seven hours, the cream pie would have approximately 768 S. aureus cells after 7 hours with a generation time of 60 minutes.
How many cells would be in the cream pie after 7 hours?Six S. aureus cells have been accidentally inoculated into a cream pie. S. aureus has a generation time of 60 minutes. S. aureus is a pathogenic bacterium found in the environment, as well as on the skin, and in the upper respiratory tract.
The generation time of this bacterium is 60 minutes, meaning that a single bacterium can produce two new cells in 60 minutes.
If there are 6 S. aureus cells in a cream pie, the number of bacteria will continue to increase as time passes.
The number of generations (n) in seven hours is calculated as:
n = t/g
n = 7 hours × 60 minutes/hour/60 minutes/generation = 7 generations
The number of cells in the cream pie after 7 hours is calculated as :
N = N₀ × 2ⁿ
N = 6 cells × 2⁷
N = 768 cells
Therefore, after seven hours, the cream pie would have approximately 768 S. aureus cells.
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What x-values are a solution to the System?
Select EACH correct answer.
Question 2 options:
x = -2
x = -1
x = 0
x = 1
x = 2
The solutions to the system equations are x = -1 and x = 0.
What is a system of equations?When two or more variables are related to one another and equations are constructed to determine each variable's value, the result is a system of equations. an equation is a balance scale, and for the equation to stay true, both sides must be equal.
Given equations are:
[tex]y=2^x - 1\\y=\frac{1}{2} x[/tex]
The solutions of the system for which; [tex]y_{1} =y_{2}[/tex]
According to the given table, the solutions to the system of both equations at x = -1 and x = 0.
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150,000 bonds with a coupon rate of 11 percent and a current price quote of 108; the bonds have 20 years to maturity. 320,000 zero coupon bonds with a price quote of 16 and 30 years until maturity. Both bonds have a par value of $1,000 and semiannual coupons
The total value of both bonds is $704,367,500.
Coupon payment = [tex]\frac{Coupon rate * Par value}{2}[/tex]
Coupon payment = [tex]\frac{11 * $1,000}{2}[/tex]
Coupon payment = $55
PV = [tex]55 * [1 - (1 + 0.04)^{^-40} ] / 0.04 + $1,000 / (1 + 0.04)^40[/tex]
[tex]PV = $1,026.45[/tex]
[tex]Total value = PV * Number of bonds * Par value\\Total value = $1,026.45 * 150,000 * $1,000\\Total value = $153,967,500[/tex]
[tex]PV = \frac{Price}{(1 + r)^n}[/tex]
[tex]PV = \frac{16}{(1 +0.03)^60}\\PV = $1.72[/tex]
[tex]Total value = PV * Number of bonds * Par value\\Total value = $1.72 * 320,000 * $1,000\\Total value = $550,400,000[/tex]
Therefore, the total value of both bonds is:
[tex]Total value = Value of coupon bonds + Value of zero coupon bonds\\Total value = $153,967,500 + $550,400,000\\Total value = $704,367,500[/tex]
A coupon rate is the annual interest rate paid by a bond or other fixed-income security to its bondholders or investors. It is typically expressed as a percentage of the bond's face value, also known as its par value. For example, if a bond has a face value of $1,000 and a coupon rate of 5%, the bond will pay $50 in interest each year to its bondholders. The coupon payments are usually made semi-annually or annually, depending on the terms of the bond.
The coupon rate is set when the bond is issued and remains fixed throughout the life of the bond unless the bond issuer chooses to call the bond or the bond defaults. Coupon rates are determined by a variety of factors, including market conditions, the creditworthiness of the issuer, and the length of the bond's maturity.
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Complete Question: -
The IPO Investment Bank has the following financing outstanding,
Debt: 150,000 bonds with a coupon rate of 11 percent and a current price quote of 108; the bonds have 20 years to maturity. 320,000 zero coupon bonds with a price quote of 16 and 30 years until maturity. Both bonds have a par value of $1,000 and semiannual coupons.
Preferred stock: 240,000 shares of 9 percent preferred stock with a current price of $67, and a par value of $100.
Common stock: 3,500,000 shares of common stock; the current price is $53, and the beta of the stock is.9.
Market: The corporate tax rate is 24 percent, the market risk premium is 8 percent, and the risk-free rate is 5 percent.
What is the WACC for the company? (Do not round intermediate calculations and enter your answer as a percent rounded to 2 decimal places, e.g., 32.16.)
seven less than the product of a number n and 1/4 is no more than 95
[tex]\frac{1}{4} n - 7 \leq 95[/tex]
A 10 kilogram object suspended from the end of a vertically hanging spring stretches the spring 9.8 centimeters. At time t = 0, the resulting mass-spring system is disturbed from its rest state by the force F(t) = 100cos(8t). The force F(t) is expressed in Newtons and is positive in the downward direction, and time is measured in seconds.
Determine the spring constant k.
k = ? Newtons / meter
Formulate the initial value problem for y(t), where y(t) is the displacement of the object from its equilibrium rest state, measured positive in the downward direction. (Give your answer in terms of y,y′,y′′,t.)
Differential equation: ?
Initial conditions: y(0) = ? and y′(0) = ?
Solve the initial value problem for y(t).
y(t) = ?
Plot the solution and determine the maximum excursion from equilibrium made by the object on the time interval 0 ≤ t < [infinity]. If there is no such maximum, enter NONE.
maximum excursion = ? meters
Using Hooke's law the maximum value of |cos(8t)| is 1, so the maximum excursion is 1/6 meters.
To find the spring constant k, we use Hooke's law:
F = -ky
where F is the weight of the object, and y is the distance it is stretched from its rest position. At equilibrium, F = mg = 10 × 9.81 = 98.1 N. Thus,
98.1 = -k × 0.098
k = -1000 N/m
The equation of motion for the system is given by:
my'' + ky = F(t)
Substituting the given values, we get:
10y'' + (-1000)y = 100cos(8t)
y'' - 100y = 10cos(8t)
with initial conditions y(0) = 0 and y'(0) = 0.
The characteristic equation is r² - 100 = 0, with roots r = ±10i. The complementary solution is therefore y_c(t) = c1cos(10t) + c2sin(10t).
For the particular solution, we assume a form of yp(t) = Acos(8t) + Bsin(8t), and substitute it in the differential equation to get:
-64Acos(8t) - 64Bsin(8t) - 100(Acos(8t) + Bsin(8t)) = 10cos(8t)
Solving for A and B, we get A = -1/6 and B = 0. Thus, the particular solution is yp(t) = (-1/6) × cos(8t).
The general solution is therefore y(t) = c1cos(10t) + c2sin(10t) - (1/6)*cos(8t). Applying the initial conditions, we get c1 = 0 and c2 = 0, so the particular solution is simply y(t) = (-1/6) × cos(8t).
The maximum excursion from equilibrium can be found by taking the absolute value of y(t) and finding its maximum value. We have:
|y(t)| = (1/6) × |cos(8t)|
The maximum value of |cos(8t)| is 1, so the maximum excursion is 1/6 meters.
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Find the equation of the straight line passing through the point (0,2) which is perpendicular to the line y=1/4x+5
Answer:
y = -4x + 2
Step-by-step explanation:
you need to find the gradient first and in order to find it, you need to look at the equation of the line given
in the equation, it refers to y = mx + c and from there, the gradient is whatever the value of m is. So in this situation, m = 1/4
now that you've found your gradient, you need to get the gradient when it is perpendicular (as stated in the question) by using m¹ x m² = -1
m¹ represents the gradient of the line we have whereas m² represents the gradient of the line we want so you just have to substitute 1/4 into m¹
[tex] \frac{1}{4} \times {m}^{2} = - 1[/tex]
[tex] {m}^{2} = \frac{ - 1}{( \frac{1}{4} )} [/tex]
[tex] {m}^{2} = - 4[/tex]
now you need to find the c of the y = mx + c before you complete the equation
y = 2 (from the question)
x = 0 (from the question)
m = -4
(2) = (-4)(0) + c
2 = c
c = 2
and you just substitute everything except y into y = mx + c and you're done
y = -4x + 2
Write an equation for an ellipse centered at the origin, which has foci at (\pm\sqrt{12},0)(± 12 ,0)left parenthesis, plus minus, square root of, 12, end square root, comma, 0, right parenthesis and vertices at (\pm\sqrt{37},0)(± 37 ,0)left parenthesis, plus minus, square root of, 37, end square root, comma, 0, right parenthesis
The equation for the ellipse is: [tex]$\frac{x^2}{37}[/tex] + [tex]$\frac{y^2}{25}[/tex] [tex]= 1$$[/tex]
The standard equation for an ellipse centered at the origin is:
[tex]$\frac{x^2}{a^2}[/tex] + [tex]$\frac{y^2}{b^2}[/tex] [tex]= 1$$[/tex]
where a is the distance from the center to a vertex, and b is the distance from the center to a co-vertex.
In this case, the vertices are located at [tex]$(\pm\sqrt{37}, 0)$[/tex], which means [tex]$a=\sqrt{37}$[/tex]. The distance between the foci is [tex]$2c=2\sqrt{12}=2\sqrt{3\times 4}=2\sqrt{3}\times 2=4\sqrt{3}$[/tex], which means [tex]$c=2\sqrt{3}$[/tex].
The value of b can be found using the relationship between a, b, and c in an ellipse:
[tex]$$a^2 = b^2 + c^2$$[/tex]
Substituting the values we know, we get:
[tex]$$37 = b^2 + (2\sqrt{3})^2$$[/tex]
Simplifying:
[tex]$$37 = b^2 + 12$$[/tex]
[tex]$$b^2 = 37 - 12 = 25$$[/tex]
Taking the square root of both sides, we get:
[tex]$$b = \pm 5$$[/tex]
Since the co-vertices are located at [tex]$(0,\pm b)$[/tex], we can see that [tex]$b=5$[/tex] (and not -5, since the ellipse is centered at the origin).
Therefore, the equation for the ellipse is:
[tex]$\frac{x^2}{37}[/tex] + [tex]$\frac{y^2}{25}[/tex] [tex]$ = 1$$[/tex]
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Can someone help me find the elevation of the sun I need the answers that are highlighted in yellow please help image below
Answer:
Step-by-step explanation:
a. ∠ACB
b. AC
c. AB
d. BC
e. tangent, opposite, adjacent
f. m∠ACB = tan⁻¹(34/45) = 37°
You pick a card at random. 1 2 3 What is P(not even)?
An 1 οr perhaps an even integer will be drawn back 75% οf the times frοm the set οf 1, 2, 3 and 4. Thus, Prοbability P(nοt even) is 75%.
Hοw simple is prοbability?Prοbability is the likelihοοd that sοmething will οccur οr the prοbability that sοmething will happen. Prοbability is the measure οf hοw prοbable it is that a cοin will land heads up after being tοssed intο the air.
As prοbabilistic arguments sοmetimes prοduce οutcοmes that appear cοntradictοry οr illοgical, prοbability is usually regarded as amοng the mοst challenging tοpics οf mathematics.
P(1) = 1/4(there is one card with a 1)
P(even) = 2/4 = 1/2 (there are 2 cards with even numbers out of 4)
Therefore,
P( 1 or even) =P(1) + P(even)
= 1/4 + 1/2
= 3/4
To express the answer as a percentage, we can multiply by 100:
P(1 or even) = 3/4 × 100%
=> 75%
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You pick a card at random. card 1, card 2 and card 3
What is P(not even)?
Write your answer as a fraction or percentage.
What is the perimeter of the trapezoid below?
Answer:
82
Step-by-step explanation:
divide this shape into a triangle and rectangle and use 24 as the base of the triangle using the Pythagoras theorem you get that the perpendicular is 7 and since 13 is also parallel to that side, 13+7=20 so one side equals 20 one side equals 25 one side equals 24 and one side equals 13 now add all of these to get 82
Step-by-step explanation:
See image below ..... perimeter = 25 + 13 + 24 + 13 + x
x is found using pythagorean theorem for right triangles
x = 7 units
total perimeter is then 82 units
1. Find the length of an arc of a circle with radius 21 m that subtends a central angle of 15°
The length of the arc is approximately 5.51 meters when a circle with a radius of 21 meters is subtended by a central angle of 15 degrees.
The length of an arc of a circle with radius 21m that subtends a central angle of 15° can be calculated using the formula:
Arc length = (central angle/360°) x 2πr
where r is the radius of the circle, and π is the mathematical constant pi.
Substituting the given values, we get:
Arc length = (15/360) x 2π x 21
Arc length = (1/24) x 2 x 3.14 x 21
Arc length = (1/12) x 3.14 x 21
Arc length = 5.51 meters (rounded to two decimal places).
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We need to apply the formula to determine the length of a circle's arc: (Central angle / 360°) x (2 x x radius) is the formula for arc length. the radius is distance from the circle center to any point on its perimeter,
and the central angle is the angle subtended by the arc at its center. The radius in this instance is stated as 21 meters, while the arc's center angle is provided as 15 degrees. When these values are added to the formula, we obtain: arc length is equal to (15°/360°) x (2x x 21m) 3.68 m. As a result, the arc measures around 3.68 meters in length. As a result, the radius is the distance from the circle's center to any point on its perimeter, if we were to sketch an arc of The arc's length would be around 3.68 meters for a circle with a radius of 21 meters and a center angle of 15 degrees.
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The monthly rent for a pizza parlor is $1,200. the average production cost per pizza is $6.75. the monthly expenses for the pizza parlor are given by the function E(x) = 1,200 + 6.75x, where x is the number of pizzas sold. for x pizzas sold, the pizza parlor's revenue is given by the function R(x) = 12.5x. the monthly profit of the pizza parlor is the difference between its revenue and its expenses. which function represents the monthly profit, P(x)?
The monthly profit, P(x), of a pizza parlor is given by the function P(x) = 5.75x - 1,200, where x is the number of pizzas sold, given revenue and expense functions.
The monthly profit, P(x), is the difference between the revenue and the expenses, so:
P(x) = R(x) - E(x)
From the given information:
R(x) = 12.5x (revenue per pizza is $12.5, and x is the number of pizzas sold)
E(x) = 1,200 + 6.75x (expenses, which include the monthly rent of $1,200 and the production cost of $6.75 per pizza, multiplied by the number of pizzas sold)
Substituting the values of R(x) and E(x) into the equation for P(x):
P(x) = 12.5x - (1,200 + 6.75x)
Simplifying the equation:
P(x) = 5.75x - 1,200
Therefore, the function representing the monthly profit, P(x), is P(x) = 5.75x - 1,200.
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Answer:
P(x)= 5.75x -1,200
Step-by-step explanation:
Plato/Edmentum
17 Troy scored 945 points playing 3 games of pinball. He scored
312 points in the first game and 356 points in the second game.
How many points did Troy score in the third game?
Answer:
Troy scored 277 points in the third game of pinball.
Step-by-step explanation:
Let x be the number of points Troy scored in the third game.
We know that Troy scored a total of 945 points in 3 games, so we can set up an equation:
312 + 356 + x = 945
Simplifying this equation, we get:
668 + x = 945
Subtracting 668 from both sides, we get:
x = 277
Therefore, Troy scored 277 points in the third game of pinball.
in a heptagon, the degree measures of the interior angles are $x, ~x, ~x-2, ~x-2, ~x 2, ~x 2$ and $x 4$ degrees. what is the degree measure of the largest interior angle?
The degree measure of the largest interior angle of the given heptagon is 132.57 degrees.
The largest interior angle of the given heptagon. The degree measures of the interior angles of a heptagon are 7, with 7 sides or vertices, are $x, ~x, ~x-2, ~x-2, ~x+2, ~x+2,$ and $x+4$ degrees.
The sum of the degree measures of the interior angles of a polygon with n sides is given by $S = (n-2) \c dot 180^\circ$. The sum of the interior angles of a heptagon is given by $S = (7-2) \cdot 180^\circ = 900^\circ$.
The sum of the degree measures of the interior angles of the heptagon is equal to $x+x+x-2+x-2+x+2+x+2+x+4 = 7x+4$. To find the value of x, we will set this equation equal to the total sum of the interior angles:$7x+4 = 900^\circ$. Solving for x, we get$x = 128.57$
We may now substitute the value of x to get the degree measures of each of the angles in the heptagon:$x = 128.57^\circ$$x = 128.57^\circ$$x - 2 = 126.57^\circ$$x - 2 = 126.57^\circ$$x + 2 = 130.57^\circ$$x + 2 = 130.57^\circ$$x + 4 = 132.57^\circ$
To find the degree measure of the largest interior angle, we must look for the angle with the largest value. We can see that the largest angle measures $132.57^\circ$.
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a garrison has provision for 30 days for certain men. if 2/3 of them do not attend the mess, then the food will last for? (b) 65 days (d) none of
A garrison has provision for 30 days for certain men. if 2/3 of them do not attend the mess, then the food will last for 45 days. So, the correct option is (a).
How to calculateGiven that the provision for certain men in the garrison is for 30 days. Also, given that 2/3 of them do not attend the mess, then we have to find the number of days the food will last.
The food will last longer if the number of people attending the mess is less because the same amount of food will have to be shared between fewer people. Therefore, the food will last for more than 30 days.
Let the total number of men be x, then the number of men attending the mess is (1/3)x
And the number of men not attending the mess is (2/3)x.
Therefore, the food will last for (30 × x) / (2/3)x = 45 days
Hence, the answer of the question is 46 days.
Your question is incomplete but most probably your full question was:
A garrison has provision for 30 days for certain men. if 2/3 of them do not attend the mess, then the food will last for?
(a) 45 days
(b) 65 days
(c) 50 days
(d) none of above
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*9. The consultancy Imagination Inc. Is working with its manufacturing client Parts-R-Us to improve their on-time
performance. The firm can earn a bonus of up to $1,000,000 based on how much the on-time performance actually
improves. It's current (baseline) on-time performance is 90%.
The company typically completes approximately 1,000 orders per month, with approximately 100 orders delayed. The
bonus payment is prorated according to the following criteria:
• The on-time performance improvement is calculated based on a reduction in late events or an improvement in on-
time performance.
• No bonus is earned for the first 25% reduction in late events, say from 100 to 75. Maximum bonus is earned once
Parts-R-Us achieves 95% on-time performance.
Please answer the following:
Write down a formula to
determine the total bonus
amount to be received
Using your formula, show
how much bonus would be
paid if Parts-R-Us achieves
94% on-time performance
Parts-R-Us could be eligible for a $0 bonus if it achieves an on-time performance rate of 94% or more.
The company would have to maintain an order delivery rate of 95% in order to qualify for the bonus. If achieved, the bonus would be as follows: Bonus = 94% * (1 - 100).
The following is the calculation for the bonus:
Bonus = P * (1 - D)
Where:
P represents the proportion of orders that are delivered on time.
D represents the total number of orders placed late.
In the event that Parts-R-Us achieves an on-time performance rate of 94%, the bonus would be as follows:
Bonus = 94% * (1 - 100)
Bonus = $0
If the company only managed to complete its tasks 94% of the time, it will not be eligible for a bonus.
In order to qualify for the maximum bonus, Parts-R-Us would have to maintain an on-time delivery rate of 95%.
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please help with question 6
Answer:
a = -13b = 6f(x) = (2x -1)(x -2)(x +3)Step-by-step explanation:
Given f(x) = 2x³ +x² +ax +b has a factor (x -2) and a remainder of 18 when divided by (x -1), you want to know a, b, and the factored form of f(x).
RemainderIf (x -2) is a factor, then the value of f(2) is zero:
f(2) = 2·2³ +2² +2a +b = 0
2a +b = -20 . . . . . . . subtract 20
If the remainder from division by (x +1) is 18, then f(-1) is 18:
f(-1) = 2·(-1)³ +(-1)² +a·(-1) +b = 18
-a +b = 19 . . . . . . . . . . add 1
Solve for a, bSubtracting the second equation from the first gives ...
(2a +b) -(-a +b) = (-20) -(19)
3a = -39
a = -13
b = 19 +a = 6
The values of 'a' and 'b' are -13 and 6, respectively.
Factored formWe can find the quadratic factor using synthetic division, given one root is x=2. The tableau for that is ...
[tex]\begin{array}{c|cccc}2&2&1&-13&6\\&&4&10&-6\\\cline{1-5}&2&5&-3&0\end{array}[/tex]
The remainder is 0, as expected, and the quadratic factor of f(x) is 2x² +5x -3. Now, we know f(x) = (x -2)(2x² +5x -3).
To factor the quadratic, we need to find factors of (2)(-3) = -6 that have a sum of 5. Those would be 6 and -1. This lets us factor the quadratic as ...
2x² +5x -3 = (2x +6)(2x -1)/2 = (x +3)(2x -1)
The factored form of f(x) is ...
f(x) = (2x -1)(x -2)(x +3)
find two positive real numbers such that the sum of the first number and the second number is 48 and their product is a maximum
Answer:
x = 24 and y = 24
Step-by-step explanation:
Let's use algebra to solve this optimization problem.
Let x be the first number, and y be the second number. Then we have the following two equations based on the problem statement:
x + y = 48 (sum of the two numbers is 48)
xy = ? (product of the two numbers, which we want to maximize)
To solve for x and y in terms of each other, we can use the fact that:
(x + y)^2 = x^2 + 2xy + y^2
Expanding the left side of the equation gives:
x^2 + 2xy + y^2 = 2304
And substituting xy for its value in terms of x and y gives:
x^2 + 2xy + y^2 = x^2 + 2(48 - x)y + y^2 = 2304
Simplifying this equation gives:
2y^2 - 96y + x^2 - 2304 = 0
To maximize the product xy, we need to maximize the value of xy = x(48 - x) = 48x - x^2. This function is a quadratic that opens downwards, and therefore, its maximum value occurs at the vertex of the parabola, which is located at x = -b/2a = -48/(2*-1) = 24.
Thus, the two positive real numbers that sum up to 48 and their product is a maximum are x = 24 and y = 24.
Emma and Cooper went to Tico’s tacos for lunch. Emma ordered three tacos and one burrito and Cooper ordered one taco and two burritos Emmas order total was $3.65 and Cooper’s bill was $3.30. Write and solve a system of equations to model the situation above. Explain the solution in the context of this problem. Explain, or show your work in the box below.
In the given system of equations one taco costs $0.80 and one burrito costs $0.72.
What is a system of equations?An equation system is a finite collection of equations for which we searched for the common solutions. It is sometimes referred to as a set of simultaneous equations or an equation set. The classification of a system of equations is similar to that of a single equation. In modelling issues where the unknown values may be expressed in the form of variables, a system of equations finds use in everyday life.
Let us suppose the cost of one taco = x.
Let us suppose the cost of one burrito = y.
Then, for Emma we have:
3x + y = 3.65
For Cooper we have:
x + 2y = 3.30
Using elimination, multiply the first equation by 2 and subtract it from the second equation:
(2)(3x + y = 3.65)
6x + 2y = 7.30
x + 2y = 3.30
-5x = -4
x = 4/5
Substituting this value of x into either equation:
3(4/5) + y = 3.65
y = 2.15/3 ≈ 0.72
Therefore, one taco costs $0.80 and one burrito costs $0.72.
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A triangle has a base that measures 6.4 feet. Its height measures 5 feet. How many square feet is the area of the triangle?
Answer:
16 feet
Step-by-step explanation:
Area of triangle = 1/2(a x b)
A = 1/2(6.4 x 5)
A = 32/2
A = 16 feet
Two similar solids have base areas of 47 cm² and 199 cm², as shown below.
The volume of the smaller solid is 350 cm³.
COMPLETION
50%
Calculate the volume of the larger solid correct to the nearest integer.
(4 marks)
Check the picture below.
so hmmm let's use the ratio for the areas to get the ratio of the sides, and from there, we'll get to the ratio of the volumes.
[tex]\stackrel{ \textit{Areas' ratio} }{\sqrt{\cfrac{s^2}{s^2}}}=\cfrac{s}{s}\implies \sqrt{\cfrac{47}{199}}=\cfrac{s}{s}\implies \cfrac{\sqrt{47}}{\sqrt{199}}=\cfrac{s}{s} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{ \textit{Volumes' ratio} }{\sqrt[3]{\cfrac{s^3}{s^3}}}=\cfrac{s}{s}\implies \stackrel{\textit{substituting from above}}{\sqrt[3]{\cfrac{s^3}{s^3}}=\cfrac{\sqrt{47}}{\sqrt{199}}}\implies \sqrt[3]{\cfrac{350}{V}}=\cfrac{\sqrt{47}}{\sqrt{199}} \\\\\\ \cfrac{350}{V}=\left( \cfrac{\sqrt{47}}{\sqrt{199}} \right)^3\implies \cfrac{350}{V}=\cfrac{\sqrt{47^3}}{\sqrt{199^3}}\implies (350)(\sqrt{199^3})=V\sqrt{47^3} \\\\\\ \cfrac{(350)(\sqrt{199^3})}{\sqrt{47^3}}=V\implies \boxed{3049\approx V}[/tex]
Kids's Kingdom, a retail toy chain, placed a seasonal order for stuffed animals from Stuffed Stuff, a distributor. Large animals cost $20, and small ones cost $14.
If the total cost of the order was $7,320 for 450 pieces, how many of each size were ordered? What was the dollar amount of each size ordered?
Answer:
Kids's Kingdom ordered 170 large stuffed animals and 280 small stuffed animals. The dollar amount of each size ordered was $3,400 for the large stuffed animals and $3,920 for the small stuffed animals.
Step-by-step explanation:
Let's use the following variables:
L for the number of large stuffed animals
S for the number of small stuffed animals
We can set up a system of two equations to represent the given information:
L + S = 450 (equation 1)
20L + 14S = 7320 (equation 2)
We can solve this system of equations using substitution or elimination. Let's use substitution.
From equation 1, we can solve for L:
L = 450 - S
Substitute this expression for L into equation 2:
20(450 - S) + 14S = 7320
Distribute the 20:
9000 - 20S + 14S = 7320
Simplify and solve for S:
6S = 1680
S = 280
So, Kids's Kingdom ordered 280 small stuffed animals. We can use equation 1 to find the number of large stuffed animals:
L + 280 = 450
L = 170
Therefore, Kids's Kingdom ordered 170 large stuffed animals.
To find the dollar amount of each size ordered, we can multiply the number of each size by the cost per item:
170 large stuffed animals at $20 each: 170 * $20 = $3,400
280 small stuffed animals at $14 each: 280 * $14 = $3,920
So, Kids's Kingdom spent $3,400 on large stuffed animals and $3,920 on small stuffed animals for a total cost of $7,320.
Consider a square whose side-length is one unit. Select any five points from inside this square. Prove that at least two of these points are within squareroot 2/2 units of each other.
The given square with a side length of one unit is known to contain five points. One must prove that at least two of these points are within square root 2/2 units of each other.
According to the Pigeonhole principle, "if n items are put into m containers, with n > m, then at least one container must contain more than one item."In this context, the square is the container, and the points inside it are the objects. If more than four points are picked, the theorem is true, and two points are nearer to each other than the square root of 2/2 units.
Let's place four points on the square's four corners. The distance between any two of these points is the square root of two units since the square's side length is 1.
Let's add another point to the mix. That point is either inside the square or outside it. Without loss of generality, let us assume that the point is inside the square. It must then be within the perimeter outlined by joining the square's corners to the point that was not a corner already.
The perimeter of the square described above is a square with a side length of square root 2 units.
Since we have five points in the square, at least two of them must be in the same smaller square, due to the pigeonhole principle. Without loss of generality, let's assume that two of the points are in the upper-left square. As a result, any points within this square are within the square root 2 units of any of the other four points. Hence, at least two points of the five selected are within the square root of 2/2 units of each other.
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