Sea "x" la longitud de cada lado del jardín original en metros.
La superficie del jardín original es x^2 m².
Si cada lado del jardín original se aumenta en 5 m, el nuevo lado del jardín será de (x+5) metros, y la nueva superficie será (x+5)^2 m².
Según el problema, la nueva superficie total es de 121 m²:
(x+5)^2 = 121
Tomando la raíz cuadrada en ambos lados:
x+5 = 11
Restando 5 en ambos lados:
x = 6
Por lo tanto, la longitud de cada lado del jardín original es de 6 metros.
Por lo tanto, la respuesta correcta es O 6 metros.
x - the length of each side of the original garden
A = x²
( x + 5 )² = 121 /√
x + 5 = 11, or x + 5 = - 11;
x = 11 - 5
x = 6 ( another solution in negative )
Answer:
A ) 6 m
Label each of the following as Discrete Random Variable, Continuous Random Variable, Categorical Random Variable, or Not a Variable. 1. The Name of the people in the car that crosses the bridge Not a Variable 2. The time between each car crossing the bridge Continuous Random Variable 3. The type of cars that cross the bridge Categorical Random Variable 4. The number of cars that use the bridge in one hour Continuous Random Variable Question 2 3 pts Which of these are Continuous and which are Discrete Random Variables? 1. Type of coin Continuous Random Variable 2. Distance from a point in space to the moon Discrete Random Variable 3. Number of coins in a stack Continuous Random Variable
Distance from a point in space to the moon is a continuous random variable and Number of coins in a stack is a discrete random variable.
A discrete random variable is one that has a finite number of possible values or one that can be countably infinitely numerous. A discrete random variable is, for instance, the result of rolling a die because there are only six possible outcomes.
A continuous random variable, on the other hand, is one that is not discrete and "may take on uncountably infinitely many values," like a spectrum of real numbers.
1. The Name of the people in the car that crosses the bridge - Not a Variable
2. Continuous random variable measuring the interval between each car crossing the bridge.
3. The Categorical Random Variable for the type of vehicles crossing the bridge
4. The number of cars that use the bridge in one hour - Continuous Random Variable
For Question 2:
1. Type of coin - Categorical Random Variable
2. The distance from a given location in space to the moon - Continuous Random Variable
3. Number of coins in a stack - Discrete Random Variable
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Complete question is:
Label each of the following as Discrete Random Variable, Continuous Random Variable, Categorical Random Variable, or Not a Variable.
1. The Name of the people in the car that crosses the bridge Not a Variable
2. The time between each car crossing the bridge Continuous Random Variable
3. The type of cars that cross the bridge Categorical Random Variable
4. The number of cars that use the bridge in one hour Continuous Random Variable
Question 2: Which of these are Continuous and which are Discrete Random Variables?
1. Type of coin
2. Distance from a point in space to the moon
3. Number of coins in a stack
The Butler family and the Phillips family each used their sprinklers last summer. The water output rate for the Butler family's sprinkler was 25 L per hour. The water output rate for the Phillips family's sprinkler was 40 L per hour. The families used their sprinklers for a combined total of 55 hours, resulting in a total water output of 1750 L. How long was each sprinkler used?
The Butler family used their sprinkler for 30 hours and the Phillips family used their sprinkler for 25 hours.
Let's solve the problem with algebra.
Let x represent the number of hours the Butlers used their sprinkler, and y represent the number of hours the Phillips family used their sprinkler. We are aware of the following:
The Butler family's sprinkler had a water output rate of 25 L per hour, so the total amount of water they used is 25x.
The Phillips family's sprinkler had a water output rate of 40 L per hour, so the total amount of water they used was 40y.
The sprinklers were used by the families for a total of 55 hours, so x + y = 55.
The total amount of water produced was 1750 L, so 25x + 40y = 1750.
Using these equations, we can now solve for x and y.
First, we can solve for one of the variables in terms of the other using the equation x + y = 55. For instance, we can solve for x as follows:
x = 55 - y
When we plug this into the second equation, we get:
25(55 - y) + 40y = 1750
We get the following results when we expand and simplify:
1375 - 25y + 40y = 1750
15y = 375
y = 25
As a result, the Phillips family ran their sprinkler for 25 hours. We get the following when we plug this into the equation x + y = 55:
x + 25 = 55
x = 30
As a result, the Butlers used their sprinkler for 30 hours.
As a result, the Butler family sprinkled for 30 hours and the Phillips family sprinkled for 25 hours.
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(b) Write 5 as a percentage.
Answer:
5 as a percentage of 100 is 5/100 which is 5%
Dividing sin^2Ø+cos^2Ø=1 by ____ yields 1+cot^2Ø=csc^2Ø
a.cot^2Ø
b.tan^2Ø
c.cos^2Ø
d.csc^2Ø
e.sec^2Ø
f.sin^2Ø
To obtain the required equation we divide the equation by sin²Ø.
What are trigonometric functions?The first six functions are trigonometric, with the domain value being the angle of a right triangle and the range being a number. The angle, expressed in degrees or radians, serves as the domain and the range of the trigonometric function (sometimes known as the "trig function") of f(x) = sin. Like with all other functions, we have the domain and range. In calculus, geometry, and algebra, trigonometric functions are often utilised.
The given equation is:
sin²Ø+cos²Ø=1
To obtain the required equation we divide the equation with sin²Ø:
sin²Ø/sin²Ø +cos²Ø/ sin²Ø = 1/sin²Ø
1 + cot²Ø = csc²Ø
Hence, to obtain the required equation we divide the equation by sin²Ø.
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find the following answer
Cardinality of given set is 10.
Describe Cardinality.The cardinality of a mathematical set refers to the number of entries in the set. It may be limited or limitless. For instance, if set A has six items, its cardinality is equivalent to 6: 1, 2, 3, 4, 5, and 6. A set's size is often referred to as the set's cardinality. The modulus sign is used to indicate it on either side of the set name, |A|.
a Set's CardinalityA set that can be counted and has a finite number of items is said to be finite. On the other hand, an infinite set is one that has an unlimited number of components and can either be countable or uncountable.
Possible set of A=14+4+1+9=28
Possible set of C=1 +6+9+9=25
n(A∩ C)=10
Hence, Cardinality of given set is 10.
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find the value of the derivative (if it exists) at
each indicated extremum.
Answer:
The value of the derivative at (0, 0) is zero.
Step-by-step explanation:
Given function:
[tex]f(x)=\dfrac{x^2}{x^2+4}[/tex]
To differentiate the given function, use the quotient rule and the power rule of differentiation.
[tex]\boxed{\begin{minipage}{5.4 cm}\underline{Quotient Rule of Differentiation}\\\\If $y=\dfrac{u}{v}$ then:\\\\$\dfrac{\text{d}y}{\text{d}x}=\dfrac{v \dfrac{\text{d}u}{\text{d}x}-u\dfrac{\text{d}v}{\text{d}x}}{v^2}$\\\end{minipage}}[/tex]
[tex]\boxed{\begin{minipage}{5.4 cm}\underline{Power Rule of Differentiation}\\\\If $y=x^n$, then $\dfrac{\text{d}y}{\text{d}x}=nx^{n-1}$\\\end{minipage}}[/tex]
[tex]\boxed{\begin{minipage}{5.4cm}\underline{Differentiating a constant}\\\\If $y=a$, then $\dfrac{\text{d}y}{\text{d}x}=0$\\\end{minipage}}[/tex]
[tex]\begin{aligned}\textsf{Let}\;u &= x^2& \implies \dfrac{\text{d}u}{\text{d}{x}} &=2 \cdot x^{(2-1)}=2x\\\\\textsf{Let}\;v &=x^2+4& \implies \dfrac{\text{d}v}{\text{d}{x}} &=2 \cdot x^{(2-1)}+0=2x\end{aligned}[/tex]
Apply the quotient rule:
[tex]\implies f'(x)=\dfrac{v \dfrac{\text{d}u}{\text{d}x}-u\dfrac{\text{d}v}{\text{d}x}}{v^2}[/tex]
[tex]\implies f'(x)=\dfrac{(x^2+4) \cdot 2x-x^2 \cdot 2x}{(x^2+4)^2}[/tex]
[tex]\implies f'(x)=\dfrac{2x(x^2+4)-2x^3}{(x^2+4)^2}[/tex]
[tex]\implies f'(x)=\dfrac{2x^3+8x-2x^3}{(x^2+4)^2}[/tex]
[tex]\implies f'(x)=\dfrac{8x}{(x^2+4)^2}[/tex]
An extremum is a point where a function has a maximum or minimum value. From inspection of the given graph, the minimum point of the function is (0, 0).
To determine the value of the derivative at the minimum point, substitute x = 0 into the differentiated function.
[tex]\begin{aligned}\implies f'(0)&=\dfrac{8(0)}{((0)^2+4)^2}\\\\&=\dfrac{0}{(0+4)^2}\\\\&=\dfrac{0}{(4)^2}\\\\&=\dfrac{0}{16}\\\\&=0 \end{aligned}[/tex]
Therefore, the value of the derivative at (0, 0) is zero.
Find the sum-of-products expansions of the Boolean function F (x, y, z) that equals 1 if and only if a) x = 0. b) xy = 0. c) x + y = 0. d) xyz = 0.
a) F(x,y,z) = y'z'. b) F(x,y,z) = x'y'z' + x'y'z + xy'z'. c) F(x,y,z) = x'y'z'. d) F(x,y,z) = x'y'z + x'yz' + xy'z' + x'y'z'. These are the sum-of-products expansions of the Boolean function F(x, y, z) for the given conditions.
a) When x = 0, F(x,y,z) equals 1 if and only if yz = 0. This can be expressed as the sum of products: F(x,y,z) = y'z' (read as "not y and not z").
b) When xy = 0, F(x,y,z) equals 1 if and only if either x = 0 or y = 0. This can be expressed as the sum of products: F(x,y,z) = x'y'z' + x'y'z + xy'z' (read as "not x and not y and not z" OR "not x and not y and z" OR "x and not y and not z").
c) When x + y = 0, F(x,y,z) equals 1 if and only if x = y = 0. This can be expressed as the sum of products: F(x,y,z) = x'y'z' (read as "not x and not y and not z").
d) When xyz = 0, F(x,y,z) equals 1 if and only if x = 0 or y = 0 or z = 0. This can be expressed as the sum of products: F(x,y,z) = x'y'z + x'yz' + xy'z' + x'y'z' (read as "not x and not y and z" OR "not x and y and not z" OR "x and not y and not z" OR "not x and not y and not z").
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One reason for using a distribution instead of the standard Normal curve to find critical values when calculating a level C confidence interval for a population mean is that
(a) z can be used only for large samples.
(b) z requires that you know the population standard deviation θ
.
(c) z requires that you can regard your data as an SRS from the population.
(d) the standard Normal table doesn't include confidence levels at the bottom.
(e) a z critical value will lead to a wider interval than a t critical value.
(b) z requires that you know the population standard deviation θ
.
Therefore , the solution of the given problem of standard deviation comes out to be the group standard deviation in order to use (b) z.
What does standard deviation actually mean?Statistics uses variance as a way to quantify difference. The image of the result is used to compute the average deviation between the collected data and the mean. Contrary to many other valid measures of variability, it includes those pieces of data on their own by comparing each number to the mean. Variations may be caused by willful mistakes, irrational expectations, or shifting economic or business conditions.
Here,
We use the z-distribution if the total standard deviation is known; otherwise, we use the t-distribution.
Additionally, for small sample sizes, the t-distribution is used, whereas for big sample sizes, the z-distribution is used.
The fact that z requires that you know the population standard deviation, and that this is frequently not known in practice, is one reason to use a distribution rather than the traditional .
Normal curve to find critical values when computing a level C confidence interval for a population mean.
You must be aware of the group standard deviation in order to use (b) z.
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please help me with math i’ll give you brainlist
Answer: False
Step-by-step explanation:
25% of the data is between Q1 and the median.
Help please! I have no idea!!!! PLEASEEE
The the inverse of [tex]n = \frac{3t+8}{5}$[/tex] is [tex]t = \frac{5n-8}{3}$[/tex].
How to find inverse of the function?To find the inverse of [tex]n = \frac{3t+8}{5}$[/tex], we need to solve for t in terms of n.
Starting with the given equation, we can first multiply both sides by 5 to get rid of the fraction:
[tex]$$5n = 3t + 8$$[/tex]
Next, we can isolate t by subtracting 8 from both sides and then dividing by 3:
[tex]$\begin{align*}5n - 8 &= 3t \\frac{5n-8}{3} &= t\end{align*}[/tex]
Therefore, the inverse of n is:
[tex]$t = \frac{5n-8}{3}$$[/tex]
We can also check that this is indeed the inverse by verifying that:
[tex]$n = \frac{3t+8}{5} = \frac{3}{5} \cdot \frac{5n-8}{3} + \frac{8}{5} = n$$[/tex]
So, the inverse of [tex]n = \frac{3t+8}{5}$[/tex] is [tex]t = \frac{5n-8}{3}$[/tex].
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0.0125 inches thick
Question 4
1 pts
The combined weight of a spool and the wire it carries is 13.6 lb. If the weight of the spool is 1.75 lb.,
what is the weight of the wire?
Question 5
1 pts
In linear equation, 11.85 pounds is the weight of the wire.
What is linear equation?
A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept. The variables in the previous sentence, y and x, are referred to as a "linear equation with two variables" at times.
Total weight of pool having 16 wires =13.6 pounds
Weight of the pool =1.75
Therefore the weight of the wire alone = 13.6 - 1.75
= 11.85 pounds
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Question 15 (2 points)
A standard deck of cards contains 4 suits of the same 13 cards. The contents of a
standard deck are shown below:
Standard deck of 52 cards
4 suits (CLUBS SPADES, HEARTS, DIAMONDS)
13 CLUBS
13 SPADES
13 HEARTS
DIAMONDS
If a card is drawn at random from the deck, what is the probability it is a jack or ten?
0
4/52- 1/13
8/52 = 2/13
48/52- 12/13
Answer: 2/13
Step-by-step explanation:
There are four jacks and four tens in a standard deck of 52 cards. However, the jack of spades and the ten of spades are counted twice since they are both a jack and a ten. Therefore, there are 8 cards that are either a jack or a ten, and the probability of drawing one of these cards at random is:
P(Jack or Ten) = 8/52 = 2/13
So the answer is 2/13.
Step-by-step explanation:
a probability is airways the ratio
desired cases / totally possible cases
in each of the 4 suits there is one Jack and one 10.
that means in the whole deck of cards we have
4×2 = 8 desired cases.
the totally possible cases are the whole deck = 52.
so, the probability to draw a Jack or a Ten is
8/52 = 2/13
Find the standard normal area for each of the following(round your answers to 4 decimal places)
Answer:
(a) 0.0955
(b) 0.0214
(c) 0.9545
(d) 0.3085
Step-by-step explanation:
You want the area under the standard normal PDF curve for intervals (1.22, 2.15), (2.00, 3.00), (-2.00, 2.00), and (0.50, ∞).
CalculatorThe probability functions of a suitable calculator or spreadsheet can find these values for you. The attachment shows one such calculator. Its "normalcdf" function takes as arguments the lower bound and upper bound.
We used 1E99 as a stand-in for "infinity" as recommended by the calculator's user manual. For the purpose here, any value greater than 10 will suffice.
any point on the parabola can be labeled (x,y), as shown. a parabola goes through (negative 3, 3)
The correct standard form of the equation of the parabola is:
[tex]y = -x^2 - 1[/tex].
To find the standard form of the equation of the parabola that passes through the given points (-3, 3) and (1, -1), we can use the general form of the equation of a parabola:
[tex]y = ax^2 + bx + c[/tex] ___________(1)
Substituting the coordinates of the two given points into this equation, we get a system of two equations in three unknowns (a, b, and c):
[tex]3 = 9a - 3b + c[/tex]
[tex]-1 = a + b + c[/tex]
To solve for a, b, and c, we can eliminate one of the variables using subtraction or addition. Subtracting the second equation from the first, we get:
[tex]4 = 8a - 4b[/tex]
Simplifying this equation, we get:
[tex]2 = 4a - 2b[/tex]
Dividing both sides by 2, we get:
[tex]1 = 2a - b[/tex]___________(2)
Now we can substitute this expression for b into one of the earlier equations to eliminate b. Using the first equation, we get:
[tex]3 = 9a - 3(2a - 1) + c[/tex]
Simplifying this equation, we get:
[tex]3 = 6a + c + 3[/tex]
Subtracting 3 from both sides, we get:
[tex]0 = 6a + c[/tex]
Solving for c, we get:
c = -6a __________(3)
Substituting this expression for c into the second equation, we get:
[tex]-1 = a + (2a - 1) - 6a[/tex]
Simplifying this equation, we get:
[tex]-1 = -3a - 1[/tex]
Adding 1 to both sides, we get:
[tex]-3a =0[/tex]
Solving for a, we get:
[tex]a = 0[/tex]
Substituting this value of a into the equation(3) for c, we get:
c = 0
Substituting a = 0 into the equation(2) for b that we found earlier, we get:
[tex]1 = 0 - b[/tex]
Solving for b, we get:
[tex]b = -1[/tex]
Putting the values of a, b and c in (1), we get
[tex]y = -x^2 - 1[/tex]
Therefore, the equation of the parabola that passes through the given points (-3, 3) and (1, -1) is:
[tex]y = -x^2 - 1[/tex]
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Complete question:
A parabola goes through (-3, 3) & (1, -1). A point is below the parabola at (-3, 2). A line above the parabola goes through (-3, 4) & (0, 4). A point on the parabola is labeled (x, y).
What is the correct standard form of the equation of the parabola?
The figure is in the image attached below
what is the number of real solutions
-X^2-9=6x
Answer options
1. Cannot be determined
2. No real solutions
3. One solution
4. two solutions
Answer:
3. One solution
Step-by-step explanation:
-x²-9 = 6x
or, x²+6x+9 = 0
or, x²+2.x.3+3² = 0 [using (a+b)² = a²+2ab+b²]
or, (x+3)² = 0
or, x+3 = 0
x = -3
4.33. Find the moment-generating function of the continuous random variable
X
whose probability density is given by
f(x)={ 1
0
for 0
elsewhere
and use it to find
μ 1
′
,μ 2
′
, and
σ 2
.
The moment-generating function of the continuous random variable X whose probability density is given by f(x) = 1 for 0 elsewhere is M(t) = 1, and its first and second central moments, μ1′ and μ2′, and the variance, σ2, are 0, 0 and 0 respectively.
The moment-generating function of the continuous random variable X whose probability density is given by f(x) = 1 for 0 elsewhere is M(t) = 1.
Using M(t) we can calculate the first and second central moments, μ1′ and μ2′, and the variance, σ2, as follows:
μ1′ = M′(t) = 0
μ2′ = M′′(t) = 0
σ2 = μ2′ - (μ1′)2 = 0 - (0)2 = 0.
Therefore, the first and second central moments, μ1′ and μ2′, and the variance, σ2, of the continuous random variable X with probability density f(x) = 1 for 0 elsewhere are 0, 0 and 0 respectively.
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Last week, the price of bananas at a grocery store was $1.40 per pound. This week, bananas at the
same grocery store are on a sale at a 10% discount. What is the total price of 6 pounds of bananas this
week at the grocery store?
A. $8.19
B. $9.18
C. $9.10
D. $8.40
According to one meaning of the phrase, it merely refers to the selling price of something. For instance, a piece of art would be sold for that amount if bids reached a record high of $10 million. Thus, option A is correct
What is the sale price by the number of pounds?The sale price of bananas this week is 10% off the original price of $1.40 per pound, which means the sale price is:
$1.40 - 10% of $1.40 = $1.26 per pound
To find the total cost of 6 pounds of bananas this week, we can multiply the sale price by the number of pounds:
$1.26 per pound * 6 pounds = $ [tex]7.56[/tex]
Therefore, the total price of 6 pounds of bananas this week at the grocery store is $ [tex]7.56[/tex] .
However, we need to be careful with the answer choices provided. They all differ from $7.56, so we need to double-check our calculations.
If we add a 10% discount to $1.40 per pound, we get:
$1.40 - (10/100)*$1.40 = $1.26 per pound
And the total cost of 6 pounds at $1.26 per pound is:
$ [tex]1.26 \times 6[/tex] = $ [tex]7.56[/tex]
Therefore, $ [tex]8.19[/tex] is not a possible answer, and the other options are either miscalculated or rounded incorrectly.
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Traffic signs are regulated by the Manual on Uniform Traffic Control Devices (MUTCD). The perimeter of a rectangular traffic sign is 126 inches. Also, its length is 9 inches longer than its widthFind the dimensions of this sign.
Answer:
Traffic signs are regulated by the Manual on Uniform Traffic Control Devices (MUTCD). The perimeter of a rectangular traffic sign is 126 inches. Also, its length is 9 inches longer than its widthFind the dimensions of this sign.
Step-by-step explanation:
Let's say the width of the sign is x inches. Then, according to the problem, the length of the sign is 9 inches longer than the width, which means the length is x + 9 inches.
The perimeter of a rectangle can be found by adding up the length of all its sides. For this sign, the perimeter is given as 126 inches. So we can set up an equation:
2(length + width) = 126
Substituting the expressions for length and width in terms of x, we get:
2(x + x + 9) = 126
Simplifying and solving for x:
2(2x + 9) = 126
4x + 18 = 126
4x = 108
x = 27
So the width of the sign is 27 inches, and the length is 9 inches longer, or 36 inches. Therefore, the dimensions of the sign are 27 inches by 36 inches.
A standard glass of wine is 5 oz. How many full glasses of wine can you get from a typical 750 ml bottle
Answer: 5 standard glasses
A standard glass of wine is 5 ounces. This is typical for any dry white, red, orange, or rosé wine. A standard bottle is 750mL, or about 25 ounces of wine. So, a normal 750mL bottle has 5 standard glasses of wine.
30 POINTS! PLEASEHELP
Answer:
Required length is 13 feet
Step-by-step explanation:
[tex]{ \rm{length = \sqrt{ {12}^{2} + {5}^{2} } }} \\ \\ { \rm{length = \sqrt{144 + 25} }} \\ \\ { \rm{length = \sqrt{169} }} \\ \\ { \rm{length = 13 \: feet}}[/tex]
Can someone help quick i have 6 questions left
Answer:
Step-by-step explanation:
long leg = 78 (means that 26√3*√3 = 26√9 = 26*3 = 78
for x: Short leg= 26√3
Hypotenuse= 2*26√3 = 52√3 for y
Let sinθ= 2√2/5 and π/2 < θ < π Part A: Determine the exact value of cos 2θ. Part B: Determine the exact value of sin (θ/2)
Answer:
Part A: To determine the exact value of cos 2θ, we can use the double-angle identity for cosine:
cos 2θ = 2 cos^2 θ - 1
We already know sin θ, so we can use the Pythagorean identity to find cos θ:
cos^2 θ = 1 - sin^2 θ
cos^2 θ = 1 - (2√2/5)^2
cos^2 θ = 1 - 8/25
cos^2 θ = 17/25
cos θ = ± √(17/25)
cos θ = ± (1/5) √17
Since θ is in the third quadrant (π/2 < θ < π), cos θ is negative, so we take the negative root:
cos θ = -(1/5) √17
Substituting into the double-angle identity:
cos 2θ = 2 cos^2 θ - 1
cos 2θ = 2 [-(1/5) √17]^2 - 1
cos 2θ = 2 (1/25) (17) - 1
cos 2θ = 34/25 - 1
cos 2θ = 9/25
Therefore, the exact value of cos 2θ is 9/25.
Part B: To determine the exact value of sin (θ/2), we can use the half-angle identity for sine:
sin (θ/2) = ± √[(1 - cos θ)/2]
We already know cos θ, so we can substitute it in:
cos θ = -(1/5) √17
sin (θ/2) = ± √[(1 - cos θ)/2]
sin (θ/2) = ± √[(1 - (-1/5) √17)/2]
sin (θ/2) = ± √[(5 + √17)/10]
sin (θ/2) = ± (1/2) √(5 + √17)
Since θ is in the third quadrant (π/2 < θ < π), sin θ is negative, so we take the negative root:
sin (θ/2) = -(1/2) √(5 + √17)
Therefore, the exact value of sin (θ/2) is -(1/2) √(5 + √17).
The exact values of the sine and cosine given are -(1/2) √(5 + √17) and 9/25.
What is the sine and the cosine of an angle?The sine of an angle in a right triangle is the ratio of the hypotenuse to the side opposite the angle.
The cosine of an angle in a right triangle is the ratio of the hypotenuse to the side adjacent the angle.
Part A: To determine the exact value of cos 2θ, we can use the double-angle identity for cosine:
cos 2θ = 2 cos² θ - 1
Using the Pythagorean identity to find cos θ:
cos² θ = 1 - sin² θ
cos² θ = 1 - (2√2/5)²
cos² θ = 1 - 8/25
cos² θ = 17/25
cos θ = ± √(17/25)
cos θ = ± (1/5) √17
Since θ is in the third quadrant (π/2 < θ < π), cos θ is negative, so we take the negative root:
cos θ = -(1/5) √17
Substituting into the double-angle identity:
cos 2θ = 2 cos² θ - 1
cos 2θ = 2 [-(1/5) √17]² - 1
cos 2θ = 2 (1/25) (17) - 1
cos 2θ = 34/25 - 1
cos 2θ = 9/25
Therefore, the exact value of cos 2θ is 9/25.
Part B: To determine the exact value of sin (θ/2), we can use the half-angle identity for sine:
sin (θ/2) = ± √[(1 - cos θ)/2]
We already know cos θ, so we can substitute it in:
cos θ = -(1/5) √17
sin (θ/2) = ± √[(1 - cos θ)/2]
sin (θ/2) = ± √[(1 - (-1/5) √17)/2]
sin (θ/2) = ± √[(5 + √17)/10]
sin (θ/2) = ± (1/2) √(5 + √17)
Since θ is in the third quadrant (π/2 < θ < π), sin θ is negative, so we take the negative root:
sin (θ/2) = -(1/2) √(5 + √17)
Therefore, the exact value of sin (θ/2) is -(1/2) √(5 + √17).
Hence, the exact values of the sine and cosine given are -(1/2) √(5 + √17) and 9/25.
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What is the equation of the line graphed?
The equation of given line which is graphed is [tex]x+2=0.[/tex] By locating the slope (m) and y-intercept (b) in the graph of a line, we can define a linear function in the form y=mx+b.
What is the formula for a line on a line graph?A straight line's graph equation can be expressed as [tex]y = m x + c[/tex] , which consists of a term, a term, and a number. a new. to the a and the likes in the likes thes of the likes of thes of thes of thes of thes of thes of people.
A line graph is a type of graph that uses straight lines to connect the data points. A line graph can show how something changes over time or compares different situations1.
A horizontal line has the equation \(y = c\), where \(c\) is a constant. This means that the \(y\)-value of every point on the line is the same
Therefore, The set of all points (x,y) in the plane that satisfy the equation [tex]y=f(x) y = f (x)[/tex] is the function's graph.
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The Khan Shatyr Entertainment Center in Kazakhstan is the largest tent in the world. The spire on top is 60 m in length. The distance from the center of the tent to the outer edge is 97.5 m. The angle between the ground and the side of the tent is 42.7°.
Find the total height of the tent (h), including the spire.
Find the length of the side of the tent (x)
i. The total height of the tent including the spire is 150 m.
ii. The length of the side of the tent x is 132.7 m.
What is a trigonometric function?Trigonometric functions are required functions in determining either the unknown angle of length of the sides of a triangle.
Considering the given question, we have;
a. To determine the total height of the tent, let its height from the ground to the top of the tent be represented by x. Then:
Tan θ = opposite/ adjacent
Tan 42.7 = h/ 97.5
h = 0.9228*97.5
= 89.97
h = 90 m
The total height of the tent including the spire = 90 + 60
= 150 m
b. To determine the length of the side of the tent x, we have:
Cos θ = adjacent/ hypotenuse
Cos 42.7 = 97.5/ x
x = 97.5/ 0.7349
= 132.67
The length of the side of the tent x is 132.7 m.
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Question 11 (1 point)
(06.03 LC)
What is the product of the expression, 5x(x2)?
a
25x2
b
10x
c
5x3
d
5x2
The expressiοn 5x(x²) is equal tο 5 * x¹ * x² (5 * x³ = 5x³). Thus, οptiοn (c) 5x3 is the cοrrect respοnse.
Hοw are prοducts οf expressiοn determined?The cοefficients (the numbers in frοnt οf the variables) οf the expressiοn 5x(x²) can be multiplied, and the expοnents οf the variables can be added, tο determine the prοduct.
The first cοefficient we have is 5 times 1, giving us 5. Sο, using the secοnd x², we have x tο the pοwer οf 2 multiplied by x tο the pοwer οf 1 (frοm the first x). Expοnents are added when variables with the same base are multiplied. Sο, x¹ multiplied by x² results in x³.
Cοmbining all οf the parts, the phrase becοmes:
5x(x²) is equal tο 5 * x¹ * x² (5 * x³ = 5x³).
Thus, οptiοn (c) 5x³ is the cοrrect respοnse.
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13/16+2 1/12+2 3/24
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Answer:
Fraction: 241/48
Improper fraction: 5 1/48
Decimal: 5.021
Step-by-step explanation:
To calculate the expression 13/16+2 1/12+2 3/24, I first converted the mixed numbers 2 1/12 and 2 3/24 to improper fractions. 2 1/12 is equal to 25/12 and 2 3/24 is equal to 51/24. Then, I added the three fractions 13/16, 25/12, and 51/24 by finding a common denominator, which in this case is 48. So, the expression becomes (39/48)+(100/48)+(102/48), which simplifies to (39+100+102)/48, which equals 241/48. Finally, I converted the improper fraction 241/48 to a mixed number, which is equal to 5 1/48.
How many different strings of length 12 containing exactly five a's can be chosen over the following alphabets? (a) The alphabet {a,b) (b) The alphabet {a,b,c}
There are 792 strings across a,b, and 27,720 in a,b,c.
(a) We must select five slots for a's in an alphabet of "a,b" before filling the remaining spaces with "b's." Hence, the binomial coefficient is what determines how many strings of length 12 that include precisely five as:
C(12,5) = 792
As a result, there are 792 distinct strings of length 12 that include exactly five a's across the letters a, b.
(b) We may use the same method as before for an alphabet consisting of the letters "a,b,c." The first five slots must be filled with a's, followed by three b's, and the final four positions must be filled with c's. The number of strings of length 12 that contain exactly five a's across the letters "a," "b," and "c" is thus given by:
C(12,5) * C(7,3) = 792 * 35 = 27720
Thus, there are 27,720 distinct strings.
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Work out the size of angle x. 79°) 35
Answer: 66
Step-by-step explanation:
all 3 of them should equal to 180
so 79+35 is 114
180-114 will give us the answer which is 66
Question 9 (2 points)
A survey asked 1,000 people if they invested in Stocks or Bonds for retirement. 700
said they invested in stocks, 400 said bonds, and 300 said both.
How many invested in neither stocks nor bonds?
Note: consider making a Venn Diagram to help solve this problem.
0
200
400
100
200 people invested in neither stocks nor bonds for retirement.
What is inclusion-exclusion principle?The inclusion-exclusion principle is a counting method used to determine the size of a set created by joining two or more sets. It is predicated on the notion that if we just sum the set sizes, we can wind up counting certain components more than once (the elements that are in the intersection of the sets). We deduct the sizes of the sets' intersections from the sum of their sizes to prevent double counting.
The total number of people who invested in stocks are:
Total = Stocks + Bonds - Both
Total = 700 + 400 - 300
Total = 800
Using the inclusion- exclusion principle:
neither = Total surveyed - Total
neither = 1000 - 800 = 200
Hence, 200 people invested in neither stocks nor bonds for retirement.
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The lunch special at Maria's Restaurant is a sandwich and a drink. There are 2 sandwiches and 5 drinks to choose from. How many lunch specials are possible?
Answer:
the question is incomplete, so I looked for similar questions:
There are 3 sandwiches, 4 drinks, and 2 desserts to choose from.
the answer = 3 x 4 x 2 = 24 possible combinations
Explanation:
for every sandwich that we choose, we have 4 options of drinks and 2 options of desserts = 1 x 4 x 2 = 8 different options per type of sandwich
since there are 3 types of sandwiches, the total options for lunch specials = 8 x 3 = 24
If the numbers are different, all we need to do is multiply them. E.g. if instead of 3 sandwiches there were 5 and 3 desserts instead of 2, the total combinations = 5 x 4 x 3 = 60.
For this question's answer, there are 2 x 5 = 10 lunch specials are possible.
The number of lunch specials possible are 10.
How many ways k things out of m different things (m ≥ k) can be chosen if order of the chosen things doesn't matter?We can use combinations for this case,
Total number of distinguishable things is m.
Out of those m things, k things are to be chosen such that their order doesn't matter.
This can be done in total of
[tex]^mC_k = \dfrac{m!}{k! \times (m-k)!} ways.[/tex]
If the order matters, then each of those choice of k distinct items would be permuted k! times.
So, total number of choices in that case would be:
[tex]^mP_k = k! \times ^mC_k = k! \times \dfrac{m!}{k! \times (m-k)!} = \dfrac{m!}{ (m-k)!}\\\\^mP_k = \dfrac{m!}{ (m-k)!}[/tex]
This is called permutation of k items chosen out of m items (all distinct).
We are given that;
Number of sandwiches=2
Number of drinks=5
Now,
To find the total number of lunch specials, we need to multiply the number of choices for sandwiches by the number of choices for drinks.
Number of sandwich choices = 2
Number of drink choices = 5
Total number of lunch specials = 2 x 5 = 10
Therefore, by combinations and permutations there are 10 possible lunch specials.
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