Solve the system of inequalities by graphing.
y≤10x–3
y>
–
1
2
x+1

The horizontal distance required for the ramp is 143.5 inches

Consider the following figure.

In right triangle ACB, CB is the horizontal distance,

hypotenuse AB = 144 in.

and the vertical distance AC = 12 in.

We need to find the measure of side CB

Let m be the measure of the horizontal distance CB

Using Pythagoras theorem for right triangle ACB,

(AB)² = (AC)² + (CB)²

Substitute given values in above equation.

(144)² = (12)² + (m)²

20736 = 144 + m^2

20736 - 144 = 144 + m^2 - 144

20592 = m^2

m^2 = 20592

m = 143.5 in.

Therefore,  the horizontal distance is 143.5 in.

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The exponential function formula for the parent function is found to be

[tex]f(x) = 2^x[/tex]

What is an exponential function?

The formula for an exponential function is [tex]f(x) = a^x[/tex], where x is a variable and a is a constant that serves as the function's base and must be bigger than 0.

The general formula to find an exponential function is [tex]f(x) = ab^x[/tex]

From the given graph, we take two points on the graph : (1,2) and (2,4)

Substituting in the above equation, taking f(x) as y

[tex]ab^1 = 2\\\\ab^2 = 4[/tex]

Solving the above equations, we get

a= 1, b=2

So the parent function  [tex]f(x) = 2^x[/tex].

Now using this function we can graph [tex]f(2x), -3f(x), 2f(2x)\\[/tex].

[tex]f(2x) = 2^{2x} = 4^x\\\\-3f(x) = -3 * 2^x\\2f(2x) = 2 * 4^x[/tex]

Therefore using the Desmos, the above exponential functions are graphed.

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∠DAC = 28°

∠BAC = 56°

PL = 20

BD =​ 31

Bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a bisector.

An angle bisector divides an angle into equal angles. If the angle is p°, the two angles made will be (p/2)°. This angle bisector passes through the vertex of an angle.

Example: Consider an Angle ∠ABC = 90°. An angle bisector will cut it into two equal angles of 45° each.

Given, AD bisects ∠BAC

MP = 20

DC = 31

∠BAD = 28°

AD bisects ∠BAC

⇒ ∠BAD = ∠DAC

⇒ ∠DAC = 28°

∠BAC = ∠BAD + ∠DAC

∠BAC = 28° + 28°

∠BAC = 56°

Since, In a figure, sides opposite to the equal angles are also equal

∠BAD = ∠DAC

⇒ BD = DC

⇒ BD = 31

⇒ BD = 31

By the figure,

∠MAP = ∠BAD and ∠PAL = ∠DAC

⇒ ∠MAP = ∠PAL

⇒ MP = PL

⇒ PL = 20

Hence, by the above calculation the measure of ∠DAC = 28°,

∠BAC = 56°, PL = 20, BD = 31.

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The expected number of ball pens of a student is 2.4.

What is the expected number?

In statistics and probability analysis, the expected value is calculated by multiplying each of the possible outcomes by the likelihood that each outcome will occur and then summing all of those values.

The expected number of ball pens of a student can be calculated by taking a weighted average of the number of ball pens that each group of students has.

The weight for each group is the percentage of students that are in that group.

So, we can calculate the expected number of ball pens of a student by:

(10% * 7) + (30% * 3) + (40% * 2) + (20% * 1)

0.7 + 0.9 + 0.8 + 0.2

The expected value is also known as the population means and it can be calculated by summing the product of each outcome’s probability and its corresponding value.

Here, the outcome is the number of ball pens.

Hence, the expected number of ball pens of a student is 2.4.

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Call the radius of the cylinder and sphere, r And the height  of the cylinder  = 2r

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Answer:

a) 4.2 - 2.8 = 1.4

b) 72.1 - 45.6 = 26.5

c) 82.2 - 41.3 - 29.2 = 11.7

d) 17.1 - 11.7 - 0.8 = 4.6

e) 29.2 - 12.5 - 0.9 = 15.8

f) 21 - 14 = 7

g) 71 - 37 - 18 = 16

h) 50 - 19 - 8 = 23

i) 92 - 57 - 16 = 19

j) 44 - 18 - 9 = 17

The fraction after solving and simplifying the mixed fraction 5 3/4 ÷ 1 1/2​ is 23 / 6.

A mixed fraction is a fraction that is created by fusing a fraction with a whole number. For instance, 8 1 / 2 is a mixed fraction if 8 is a whole number and 1/2 is a fraction.

Given:

5 3/4 ÷ 1 1/2​

Solve the mixed fraction into improper fractions as shown below,

5 × 4 + 3 / 4 ÷ 1 × 2 + 1 / 2

20 + 3 / 4 ÷ 2 + 1 / 2

23 / 4 ÷ 3 / 2

23 / 4 × 2 / 3

23 × 2 / 4 × 3

On simplifying,  

23 × 1 / 2 × 3

23 / 6

Thus, the value is 23 / 6.

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2a+b=15.7 is the formula used to get the base's length and b has a value of 3.

Therefore, an isosceles triangle has two equal sides as well as two equal angles. The Greek words iso (same) and Skelos are the source of the name (leg). An equilateral triangle is one in which all of its sides are equal, whereas a scalene triangle is one in which none of its sides are equal.

An isosceles triangle has 2 equal sides that are longer than the length of the base

perimiter=15.7

equation is 2a+b=15.7

so if the longer side is 6.3

we know that identical sides are bigger so 6.3 is one of the identical sides

2a means the 2 identical sides

2(6.3) + b = 15.7

12.6 + b = 15.7

subtract 12.6 from both sides

b = 3.1

therefore, The equation used to find the length of the base is 2a+b=15.7

and the value of b is 3.1

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Answer:

12.6 + b = 15.7

The measure of angle B is 107°.

What do similar triangles mean?

If two triangles have an equal number of corresponding sides and an equal number of corresponding angles, then they are comparable. Similar figures are described as items with the same shape but varying sizes, such as two or more figures.

Given a triangle, NPZ is similar to triangle HBZ.

Which can also be written as ΔNPZ ≈ ΔHBZ

So, by equating the corresponding angles that are angle P = angle B:

8x-37°=180-(2x-5+42)°

10x=180

x=18°

So, H=36-5=31°

Since the sum of all angles in a triangle is 180°, angle B can be calculated  as

B= 180°-(42+31)°

B=107°

Therefore the measure of angle B is 107 degrees

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x->∞,y->∞ and as x-> - ∞ , y-> - ∞ this statement  best describes the end behavior.

Graph is a mathematical representation of a network and it describes the relationship between lines and points.

First, we need to analyze the plotting of the graph. When it is on the negative axis of x, as the values of x increase (x values increasing in the negative axis means it's decreasing in value), the y values also increase. So as x decreases, y increases.

On the positive axis of x, as the x values increase, the corresponding values of y also increase with x. So as x increases, y values also increase with it.

Hence, x->∞,y->∞ and as x-> - ∞ , y-> - ∞ this  statement  best describes the end behavior.

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When Ava purchased six packages of tulip bulbs and twelve bags of daffodil bulbs, the price per tulip bulb was $7 and the price each daffodil bulb was $13.

The algebraic equation y=mx+b is known as a linear equation. M serves as the y-intercept, and B serves as the slope. The previous clause has two variables, y and x, and is sometimes referred to as a "linear equation with two variables". Bivariate linear equations are those with two independent variables. The following are a few examples of system of equations: 2x - 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, and 3x - y + z = 3. When an equation takes the form y=mx+b, with m denoting the slope and b denoting the y-intercept, it is referred to as being linear. The term "linear" refers to an equation having the form y=mx+b, where m stands for the slope and b for the y-intercept.

given

Let x be the price of a single tulip bag.

Y should equal the price of one bag of daffodil bulbs.

Grace = 14x + 12y = 254

ava = 6x + 12y = 198

- 8x = - 56

x = - 56/-8

x = 7.

Equation 1 becomes

6* 7 + 12y = 198

42 + 12y = 198 when x = 7 is substituted.

12y = 198 - 42

12y = 156

y = 156/12

y = 13

When Ava purchased six packages of tulip bulbs and twelve bags of daffodil bulbs, the price per tulip bulb was $7 and the price each daffodil bulb was $13.

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Answer:

f(6) = - 50

Step-by-step explanation:

substitute x = 6 into f(x) , that is

f(6) = - 6² - 14 = - 36 - 14 = - 50

Answer:

-50

Step-by-step explanation:

f ( x ) = - x² - 14

Accordingly, to find the value of f ( 6 ) you have to replace x with ( 6 ).

f ( 6 ) = - (6)² - 14

f ( 6 ) = - 36 - 14

f ( 6 ) = -50

A particle traveled a distance of 9.0×10¹² meters in 5.0 × 10⁷ seconds.

The division in mathematics is one kind of operation. In this process, we split the expressions or numbers into the same number of parts.

Given:

If a particle traveled a distance of 9.0×10¹² meters at a rate of 1.8×10⁵meters per second.

And the distance,

= rate x time.

Rearranging the formula,

Time = Distance / rate

Time = 9.0×10¹² / 1.8×10⁵

Simplifying by using division operation,

Time = (9.0/1.8) × 10¹² × 10⁻⁵

Time = 5.0 × 10⁷

Therefore, the required specified time is  5.0 × 10⁷ seconds.

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To fill one urn with one white ball and the second urn with the remaining nine white and ten black balls.

Probability measures how probable something is to occur. We can discuss the probabilities of various outcomes, or how likely they are, whenever we are unsure of how an event will turn out. Statistics is the study of events subject to probability.

By dividing the number of preferable possibilities by the total number of potential outcomes, probability, which measures the likelihood that an event will occur, is obtained. A coin flip is the most basic illustration. There are only two outcomes that can occur when you flip a coin: either heads or tails.

The basic probability formula is: Probability = Number of Favorable Outcomes / Total Number of Outcomes. or P(A) = f/N

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Answer:

It will either invert the graph or stretch/compress the horizontal

Step-by-step explanation:

When a is negative It will invert the graph, and reflect it over the x-axis

By plugging in a=1 you get a normal absolute value function of x ( a V on the graph)

If a>1 then your graph will compress

If a<1 then your graph will stretch

The algebraic expression for the given sequence 1, 10, 19 as required to be determined is; a (n) = 1 + ( n - 1 ) 9.

As evident in the task content; it follows that the common difference of the given sequence is constant since; 19 - 10 = 10 - 1 = 9.

On this note, since the nth term of arithmetic sequences take the form;

a (n) = a (1) + ( n - 1 )d

Since the first term is; 1 and the common difference is; 9; the resulting algebraic expression is;

a (n) = 1 + (n - 1)9.

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The probability of picking a card at random from a standard deck of 52 cards is 0.019 or 0.02(approx)

Probability is a measure of the likelihood of an event occurring. The probability formula is defined as the possibility of an event happening being equal to the ratio of the number of favorable outcomes and the total number of outcomes. Probability of event to happen P(E) = Number of favorable outcomes/Total Number of outcomes.

Given here: A card is drawn from a standard deck of 52 cards. Let the probability be P then P= 1/52

                                     =0.02

Hence, The probability of picking a card at random from a standard deck of 52 cards is 0.019 or 0.02(approx)

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2 is not a polynomial root of  [tex]x^{3} + 4x^{2} + 3x[/tex] . A polynomial is an equation that employs addition, subtraction .

A polynomial is a mathematical statement made up of variables (also known as indeterminates) and coefficients that uses solely the operations addition, subtraction, multiplication, and non-negative integer exponentiation of variables. A polynomial is an equation that employs addition, subtraction, multiplication, and division of numbers to combine variables, constants, and exponents (No division operation by a variable).

given

[tex]x^{3} + 4x^{2} + 3x[/tex]

= [tex]2^{3} + 2^{2} + 3(2) = 0\\8 + 4 + 6 = 0 \\18= 0[/tex]

Consequently, 2 is not a polynomial root of  [tex]x^{3} + 4x^{2} + 3x[/tex] .

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The formula for K in terms of F is [tex]K = \frac{5}{9} (F-32) + 273[/tex]

The given expression:

[tex]F = \frac{9}{5} (K-273) + 32[/tex]

To find the formula for K

The formula for K is obtained by making K the subject of the formula as shown below:

[tex]F = \frac{9}{5} (K-273) + 32[/tex]

subtract 32 from LHS.

[tex]F-32 = \frac{9}{5} (K-273)[/tex]

multiply 5 with LHS

[tex]5(F-32) = 9(K-273)[/tex]

Take numerator 9 to LHS

[tex]\frac{5}{9}(F-32) = K-273[/tex]

take -273 from RHS to LHS

[tex]\frac{5}{9}(F-32) + 273 = K[/tex]

Thus the formula for K in terms of F is [tex]K = \frac{5}{9} (F-32) + 273[/tex]

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"If p then q" is a conditional statement, and its contrapositive is If ~q then ~p.

Knowing about the inverse statement makes it simple to comprehend how to build a contrapositive statement. Take the negation of the hypothesis and the conclusion to produce the conditional statement's opposite.

Create the inverted statement first, then swap the hypothesis and conclusion to create the contrapositive of the conditional statement. Adding "not" to both component statements and switching the order of the provided conditional statements will result in contrapositive assertions, in other words.

If a number n is even, then n2 is even, and vice versa, is the contrapositive statement to "If a number n is even, then n2 is even."

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Based on the information presented, the ratio represents 8th-grade baseball players to total athletes.

A ratio is a mathematical expression that shows the proportion or the number of a specific element in a category compared to the number of another element in the same category. For example, if you count the animals on a farm the ratio of cows to chickens can be 5:7, which means that for every 5 cows there are 7 chickens.

5 : This is the number of baseball players who are in 8th-grade

73: This is the total of players. You get this number if you add all the numbers given.

Based on this, this ratio is comparing the number of baseball players and total players, which means the meaning of this ratio is " the ratio of 8th grade baseball players to total athletes".

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The score of Larry is 32 points.

Two algebraic expressions having same value and symbol '=' in between are called as an equation.

Given:

Ervin scores half as many points in a game as Larry.

If Larry scored x points,

then the points of Ervin are x/2 points.

Here, x = 16,

So,

Ervin's points = Larry's points

The equation is,

16 = x/2

x = 16(2)

x = 32 points

Therefore, 32 points are the score of Larry.

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The equation of the linear function after a translation down 9 units is given as follows:

f(x) = x - 11.

The slope-intercept definition of a linear function is given as follows:

y = mx + b.

In which:

The graph of the function crosses the y-axis at y = -2, hence the intercept b is given as follows:

b = -2.

When x increases by 2 units, from x = 0 to x = 2, y increases by 2 units, from y = -2 to y = 0, hence the slope is calculated as follows:

m = 2/2.

Then the function is of:

y = x - 2.

After the translation down 9 units, 9 is subtracted from the function, hence:

f(x) = x - 2 - 9

f(x) = x - 11.

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We can find the x-intercept by writing the equation in intercept form which  is also known as factored form, y= (x-p) (x-q) where p, q are the x-intercepts.

To find the vertex of the quadratic function we can write the function  in standard form , the vertex is located at the point where y=f (x).

A function which has a degree of two is known as a quadratic function.

Example, 3x² + 2 , here the maximum degree is 2 , hence the given function is quadratic. The general form of a quadratic equation is given as ,

f(x) = ax2 + bx + c .

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Answer:

Step-by-step explanation:

Perimeter is the addition of all the sides.

So in this question, if you add up all the sides :

14x + 8x +7x +12

Simplify by adding all terms with x :

29x + 12  

(As you can not add a number constant to a variable )

Answer: Perimeters Are Shown Below

Step-by-step explanation:

6. 2 Hexagons; 1+1+1+1+1+1+1+1+1+1   OR 1 x 10

6. 3 Hexagons; 1+1+1+1+1+1+1+1+1+1+1+1+1+1  OR 1 x 14

7. 2 Pentagons; 1+1+1+1+1+1+1+1  OR 1 x 8

7. 3 Pentagons; 1+1+1+1+1+1+1+1+1+1+1  OR 1 x 11

The minimum and maximum possible heights of the bowling pin are 14.96875 inches  and 15.03125 inches

From the question, we have the following parameters that can be used in our computation:

Absolute deviation = 0.03125 inches

Height = 15 inches

The minimum is calculated as

Minimum = Height - Absolute deviation

Substitute the known values in the above equation, so, we have the following representation

Minimum = 15 - 0.03125 inches

Minimum = 14.96875 inches

The maximum is calculated as

Maximum = Height + Absolute deviation

Substitute the known values in the above equation, so, we have the following representation

Maximum = 15 + 0.03125 inches

Minimum = 15.03125 inches

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The x-intercept of the line 4x-y =- 4  is (1,0).

At x-intercepts, y=0  and at

y-intercepts, x=0.

So, to find the,

x-intercept, set y=0

4x−0=4

4x=4

x=1.

The points at which the graph of a function or an equation crosses or "touches" the x-axis of the Cartesian Plane are referred to as the x-intercepts. This could be thought of as a point with a y-value of zero. Let y equal 0 and then solve for x to determine the equation's x-intercepts.

Because the line crosses the x-axis when y=0, the equation for x must be solved in order to determine the x-intercept. We can solve for the intercepts when an equation is not in the form y = mx + b by solving for the remaining variable and plugging in 0 as necessary. To locate the y-intercept: solve for y and set x to zero.

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The ratio of yellow paint to mixed paint is 3/5.

A ratio is a comparison or relationship between two amounts of the same quantity to determine how much larger one number is than the other. Where b is not equal to zero, it is written as a/b or a: b. An equality of two ratios is a proportion. In order to write equivalent ratios, proportions are used, which aids in resolving the unknown quantities. As an illustration, a proportion is written as a: b = c: d.

Solution:-

Given,

Amount of yellow paint=3

Amount of blue paint=2

Ratio=yellow paint/total mixed paint

Ratio=3/(3+2)

Ratio=3/5

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