100 Points! Algebra question. Sketch the angle. Then find its reference angle. Show your calculations. Photo attached. Thank you!

Int he above expression, A = 3 (distance from the vertex to the center)

B = 4

C = 3 (distance from focus to center)

D = 10

Since the center of the hyperbola is (3,10), we have C=3 and D=10.

The distance from the center to the vertex is A, so we have A= 6-3

A = 3.

The distance from the center to the focus is given by c, so we have c=8-3=5.

We can use the relationship a² + b² = c² to solve for B:

a = 6 - 3 = 3 (distance from vertex to center)

c = 5 (distance from focus to center)

b = ?

b² = c² - a²

b² = 5² - 3²

b² = 16

b = 4

Therefore, the equation of the hyperbola is

((x-3)²/3²) - ((y-10)²/4²) = 1

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Full Quesitn:

The equation of the hyperbola that has a center at (3, 10), a focus at (8, 10), and a vertex at (6, 10), is

((x-C)²/A²) - ((y-D)²)/B²) = 1

Where A = ?

B  = ?

C = ?

D = ?

Answer:

65 units

Step-by-step explanation:

solution Given:

apothem(a)=10

no of side(n)= 10

First, we need to find the length of one side (s).

We can find the length of one side using the following formula:

[tex]\boxed{\bold{s = 2 * a * tan(\frac{\pi}{n})}}[/tex]

substituting value:

[tex]\bold{s = 2 * 10 * tan(\frac{\pi}{10})=6.498}[/tex]   here π is 180°

Now

Perimeter: n*s

substituting value:

Perimeter = 10*6.498= 64.98 in nearest tenth 65 units

Therefore, the Perimeter of a regular decagon is 65 units.

Answer:

65.0 units

Step-by-step explanation:

A regular decagon is a 10-sided polygon with sides of equal length.

To find its perimeter, we first need to find its side length (s).

As we have been given its apothem, we can use the apothem formula to find an expression for side length (s).

[tex]\boxed{\begin{minipage}{5.5cm}\underline{Apothem of a regular polygon}\\\\$a=\dfrac{s}{2 \tan\left(\dfrac{180^{\circ}}{n}\right)}$\\\\where:\\\phantom{ww}$\bullet$ $s$ is the side length.\\ \phantom{ww}$\bullet$ $n$ is the number of sides.\\\end{minipage}}[/tex]

Given the apothem is 10 units and the number of sides is 10, substitute a = 10 and n = 10 into the formula and solve for s:

[tex]10=\dfrac{s}{2 \tan \left(\dfrac{180^{\circ}}{10}\right)}[/tex]

[tex]10=\dfrac{s}{2 \tan \left(18^{\circ}\right)}[/tex]

[tex]s=20 \tan \left(18^{\circ}\right)[/tex]

The perimeter (P) of a regular polygon is the product of its side length and the number of sides. Therefore, the perimeter of the given regular decagon is:

[tex]P=s \cdot n[/tex]

[tex]P=20 \tan \left(18^{\circ}\right) \cdot 10[/tex]

[tex]P=200 \tan \left(18^{\circ}\right)[/tex]

[tex]P=64.9839392...[/tex]

[tex]P=65.0\; \sf units\;(nearest\;tenth)[/tex]

Therefore, the perimeter of a regular decagon with an apothem of 10 units is 65.0 units, to the nearest tenth.

Answer: vvvvv

Step-by-step explanation:

Let R1 and G1 represent the red and green apples available for the first selection, respectively. Similarly, let R2, G2a, and G2b represent the red and two green apples available for the second selection, respectively. Then, the sample space for the two selections is:

{(R1, R2), (R1, G2a), (R1, G2b), (G1, R2), (G1, G2a), (G1, G2b)}

Out of these six outcomes, only two involve selecting a green apple first: (G1, R2) and (G1, G2a). And out of these two outcomes, only one involves selecting a green apple second: (G1, G2a).

Therefore, since the probability of selecting a green apple second changes based on whether a green apple was selected first, the events of selecting a green apple first and selecting a green apple second are dependent.

To find the number of buttons in each of Jack's five bags, we can use the information given to us and solve for the unknowns. Let's start by finding the total number of buttons:

Total number of buttons = Mean number of buttons × Number of bags

Total number of buttons = 27 × 5

Total number of buttons = 135

Since we know the fewest and greatest number of buttons in a bag, we can find the range:

Range = Greatest number of buttons - Fewest number of buttons

Range = 29 - 21

Range = 8

Now, we can set up two equations to find the two modes:

Mode 1 + Mode 2 + 3 × 27 = 135 (sum of all bags)

Mode 2 - Mode 1 = 8 (difference between the two modes)

Solving these equations simultaneously, we get:

Mode 1 = 23

Mode 2 = 31

Therefore, the number of buttons in each of Jack's five bags is:

Bag 1: 21 buttons

Bag 2: 23 buttons

Bag 3: 25 buttons

Bag 4: 27 buttons

Bag 5: 29 buttons

The transformation of Δ ABC  to ΔADE is a dilation by a scale factor of 1/2 and then 180 degrees rotation about the origin.

We have,

In ΔABC,

The coordinates of each point are:

A = (0, 0)

B = (0, -6)

C = (8, -6)

And,

ΔADE,

A = (0, 0)

D = (0, 3)

E = (4, 3)

Now,

We can see that,

If we use a scale factor 1/2 on each the coordinates of A, B, and C and rotated to 180 degrees about the origin.

We get,

A = (0, 0) = (0, 0)

B = (0, - 6) = (0, -3) = (0, 3)

C = (8, -6) = (4, -3) = (4, 3)

Thus,

The transformation of Δ ABC  to ΔADE is a dilation by a scale factor of 1/2 and then 180 degrees rotation about the origin.

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Each number should be dragged to a box to complete the table as follows;

Kilometers     Meters

1                        1,000

2                       2,000

3                       3,000

5                       5,000

8                       8,000

In Science and Mathematics, a conversion factor can be defined as a number that is used to convert a number in one set of units to another, either by dividing or multiplying.

Generally speaking, there are one (1) kilometer in one thousand (1,000) meters. This ultimately implies that, a proportion or ratio for the conversion of kilometer to meters would be written as follows;

Conversion:

1 kilometer = 1,000 meters

2 kilometer = 2,000 meters

3 kilometer = 3,000 meters

4 kilometer = 4,000 meters

5 kilometer = 5,000 meters

6 kilometer = 6,000 meters

7 kilometer = 7,000 meters

8 kilometer = 8,000 meters

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

The measure of segment HI is given as follows:

HI = 7.

The measure of segment HI for this problem are obtained considering the triangle midsegment theorem, which states that the length of the midsegment of the triangle is equals to half the length of the base, hence the base has the length that is twice the midsegment.

The lengths are given as follows:

Hence the value of x is obtained as follows:

EF = 2HI

-21 + 5x = 2(-9x + 70)

-21 + 5x = -18x + 140

23x = 161

x = 7.

Hence the length HI is given as follows:

HI = -9(7) + 70

HI = 7.

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Hello !

Answer:

[tex]\boxed{\sf (3x + 4)(x - 1)=3x^2+x-4}[/tex]

Step-by-step explanation:

We will have to use the distributive property to expand this expression.

Let's remember :

[tex]\sf (a+b)(c+d)=ab+ad+bc+bd)[/tex]

Let's apply this property to our expression :

[tex]\sf (3x + 4)(x -1)\\=3x\times x+3x\times(-1)+4\times x+4\times(-1)[/tex]

Now let's calculate and combine like terms :

[tex]\sf 3x^2-3x+4x-4\\\boxed{\sf =3x^2+x-4}[/tex]

Have a nice day ;)

We cannot conclude that there are more than 70,000 defined words in the dictionary.

To test the claim that there are more than 70,000 defined words in the dictionary, we can set up the null and alternative hypotheses as follows:

Null Hypothesis (H0): The mean number of defined words on a page is 48.0 or less.

Alternative Hypothesis (H1): The mean number of defined words on a page is greater than 48.0.

So, sample mean

= (59 + 37 + 56 + 67 + 43 + 49 + 46 + 37 + 41 + 85) / 10

= 510 / 10

= 51.0

and, the sample standard deviation (s)

= √[((59 - 51)² + (37 - 51)² + ... + (85 - 51)²) / (10 - 1)]

≈ 16.23

Next, we calculate the test statistic using the formula:

test statistic = (x - μ) / (s / √n)

In this case, μ = 48.0, s ≈ 16.23, and n = 10.

test statistic = (51.0 - 48.0) / (16.23 / √10) ≈ 1.34

With a significance level of 0.05 and 9 degrees of freedom (n - 1 = 10 - 1 = 9), the critical value is 1.833.

Since the test statistic (1.34) is not greater than the critical value (1.833), we do not have enough evidence to reject the null hypothesis.

Therefore, based on the given data, we cannot conclude that there are more than 70,000 defined words in the dictionary.

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The correct option is B, the total volume is 6 gallons, 2 quarts, and 1 pint.

Here we just need to add all the volumes that are in the question, we have.

So there are:

3 gallons.

8 quarts.

13 pints.

Now let's do the changes of units, we know that:

8 pints = 1 gallon

4 quarts = 1 gallon

1 quart = 2 pints

then:

13 pints = 1 gallon  and 5 pints

8 quarts = 2 gallons

5 pints = 2 quarts and 1 pint

Then the total volume is:

6 gallons, 2 quarts, and 1 pint.

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The values of Composite function are:

f(g(-1)) = 20

h[g(4)] = 40

g[f(5)] = -37

f[h(-4)] = -44

g[g(7)] = 35

h[f(1/2)] = -8

We have, f(x) = -2x, g(x) = 3x-7 and h(x)= 2x² -10

f(g(-1)):

First, substitute -1 into g(x): g(-1) = 3(-1) - 7 = -3 - 7 = -10.

Next, substitute the result into f(x): f(-10) = -2(-10) = 20.

Therefore, f(g(-1)) = 20.

h[g(4)]:

First, substitute 4 into g(x): g(4) = 3(4) - 7 = 12 - 7 = 5.

Next, substitute the result into h(x): h(5) = 2(5^2) - 10 = 2(25) - 10 = 50 - 10 = 40.

Therefore, h[g(4)] = 40.

g[f(5)]:

First, substitute 5 into f(x): f(5) = -2(5) = -10.

Next, substitute the result into g(x): g(-10) = 3(-10) - 7 = -30 - 7 = -37.

Therefore, g[f(5)] = -37.

f[h(-4)]:

First, substitute -4 into h(x): h(-4) = 2(-4²) - 10 = 2(16) - 10 = 32 - 10 = 22.

Next, substitute the result into f(x): f(22) = -2(22) = -44.

Therefore, f[h(-4)] = -44.

g[g(7)]:

First, substitute 7 into g(x): g(7) = 3(7) - 7 = 21 - 7 = 14.

Next, substitute the result into g(x) again: g(14) = 3(14) - 7 = 42 - 7 = 35.

Therefore, g[g(7)] = 35.

h[f(1/2)]:

First, substitute 1/2 into f(x): f(1/2) = -2(1/2) = -1.

Next, substitute the result into h(x): h(-1) = 2(-1²) - 10 = 2(1) - 10 = 2 - 10 = -8.

Therefore, h[f(1/2)] = -8.

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Answer:

4.1 = 4.10

Step-by-step explanation:

Extra 0 at the end of 4.10 doesn't change the quantity of the number, so 4.1 is equal to 4.10

Answer:

D

Step-by-step explanation:

sin  (  3pi / 4 )

 =  sin   (   pi   -   pi  /  4   )

 =  sin   (   pi / 4  )

 = 1/root(2)

  =  root(2) / 2

The three consecutive multiples of 3 are 15, 18 and 21

First, let's determine three successive multiples of 3:

The subsequent two would be "x+3" and "x+6" if we call the initial number "x".

Since we are aware that the third number (x+6) is one more than the first two numbers (x + x+3), we can write the following equation:

2/3(x + x+3) = (x+6) + 1

Simplifying this equation, we get:

2/3(2x+3) = x+7

Multiplying both sides by 3, we get:

2(2x+3) = 3(x+7)

Expanding and simplifying, we get:

4x + 6 = 3x + 21

Subtracting 3x and 6 from both sides, we get:

x = 15

Therefore, the three consecutive multiples of 3 are 15, 18 and 21

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Using  the associative law of multiplication we get this two solution of the given expression: (12x)y = (12 * x) * y , (12x)y = 12 * (x * y)

The associative law of multiplication states that when multiplying three or more numbers, the product is the same regardless of the order in which the numbers are grouped. In other words, it doesn't matter which numbers we multiply first, the result will be the same.

Using the associative law of multiplication, we can group the factors in any way we like. Therefore, we can write an equivalent expression to (12x)y by changing the grouping of factors.

One way to group the factors is to group the two numerical coefficients (12 and y) together.

Another way to group the factors is to group the variable x and the coefficient y together.

Both of these expressions are equivalent to (12x)y and they all follow the associative law of multiplication.

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The  values for the Number of cheeseburgers, fries, candy bars, and bottle sodas seem to be inconsistent.The number of fries (Y) is 122.50.The number of cheeseburgers (X) is -17.50

We can calculate the values of X and Y, representing the number of cheeseburgers and fries respectively, and the number of candy bars and bottle sodas respectively.

For cheeseburgers and fries:

We know that the expected revenue from cheeseburgers and fries is $175.

The equation relating the cost of the cheeseburgers (X) and fries (Y) is given as 4X + 2Y = 175.

We also know that the total number of cheeseburgers and fries to be sold is 105, which can be represented as X + Y = 105.

To solve for Y, we can substitute X = 105 - Y into the first equation:

4(105 - Y) + 2Y = 175

420 - 4Y + 2Y = 175

-2Y = 175 - 420

-2Y = -245

Y = -245 / -2

Y = 122.50

To solve for X, we substitute Y = 122.50 into the equation X + Y = 105:

X + 122.50 = 105

X = 105 - 122.50

X = -17.50

Therefore, the number of cheeseburgers (X) is -17.50. However, since it is not possible to have a negative quantity of cheeseburgers, we can conclude that there might be an error in the calculations or the given information.

Moving on to candy bars and bottle sodas:

We expect to make $120 from the sales of candy bars and bottle sodas.

The equation relating the cost of candy bars (X) and bottle sodas (Y) is given as 1.85X + 2.49Y = 120.

The total number of candy bars and bottle sodas to be sold is 75, which can be represented as X + Y = 75.

To solve for Y, we substitute X = 75 - Y into the first equation:

1.85(75 - Y) + 2.49Y = 120

138.75 - 1.85Y + 2.49Y = 120

0.64Y = 120 - 138.75

0.64Y = -18.75

Y = -18.75 / 0.64

Y = -29.29

Again, we encounter a negative value, which is not possible for the number of bottle sodas. This suggests a mistake in the calculations or the given information.

Similarly, to solve for X, we substitute Y = -29.29 into the equation X + Y = 75:

X - 29.29 = 75

X = 75 + 29.29

X = 104.29

Here, we obtain a non-integer value for the number of candy bars, which may indicate an error in the calculations or the given information.

In conclusion, the calculated values for the number of cheeseburgers, fries, candy bars, and bottle sodas seem to be inconsistent with the given information

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Answer:

its b4 + 7b3 + 4b2 + b5 + 7b4+ 4b3= b5+8b4+11b3+4b2

Answer:

When multiplying, add the exponents, (example) remember if there is "7b" the exponent is one.

Multiply b^2 * b^3 = b^5  (add the exponent 2 + 3 = 5)

Multiply 7b * b^3 = 7b^4 (the exponent of 7b is one, add 1 + 3 for the exponent to become 4)

Multiply 4 * b^3 = 4b^3 (4 doesn't have a variable, the exponent will be 3)

b^2 * b*2 = b^4 (add exponents)

7b * b^2 = 7b^3 (add the exponents 1 + 2)

4 * b^2 = 4b^2

              b^2             +               7b             +        4

b^3           b^5              +           7b^4            +   4b^3

+                                                                                      

b^2           b^4             +           7b^3            + 4b^2

b^5 + 7b^4 + 4b^3 + b^4 + 7b^3 + 4b^2

[tex]b^5 + 7b^4 + 4b^3 + b^4 + 7b^3 + 4b^2[/tex]

Answer:

3x + 12

Step-by-step explanation:

Using the distributive property, multiply 3 by x and get 3x. Then multiply 3 by 4 and get 12. Put it together and get 3x + 12

The events A and B are dependent events.

The probability of event B will be different from the initial probability of selecting a black sock (event A).

Two events are considered independent if the outcome of one event does not affect the probability of the other event.

The outcome of event A (choosing a black sock) directly affects the probability of event B (choosing a black sock again).

To explain further, let's consider the initial scenario:

You have 10 white socks and 14 black socks in your drawer.

The total number of socks is 24.

The first sock, there are two possibilities:

Either it is a black sock or a white sock.

If event A occurs and you select a black sock, the total number of black socks in the drawer decreases to 13, while the total number of socks decreases to 23.

If event B to occur (selecting a black sock again), the probability is now influenced by the fact that you have one less black sock and one less sock in the drawer.

The probability of event B will be different from the initial probability of selecting a black sock (event A).

The outcome of event A affects the probability of event B, the two events are dependent.

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If Eric prunes 7 trees, he would earn $85.

To write an equation that relates the number of trees pruned (x) to the amount Eric earns (y), we can use the given data points to determine the pattern or relationship.

From the table, we observe that Eric's pay increases as the number of trees pruned increases, except for the case where he prunes 8 trees.

Based on the data, we can construct a piecewise equation:

For x = 2, 4, and 6:

y = 30 * x

For x = 8:

y = 80

However, we need to consider the case when x = 7. Since this data point is missing from the table, we can assume that the pay is proportional to the number of trees pruned. Therefore, we can interpolate using the given data points for x = 6 and x = 8:

Using the formula for linear interpolation:

y = y1 + (y2 - y1) * ((x - x1) / (x2 - x1))

Substituting the values, we have:

y = 90 + (80 - 90) * ((7 - 6) / (8 - 6))

y = 90 + (-10) * (1 / 2)

y = 90 - 5

y = 85

Thus, if Eric prunes 7 trees, he would earn $85.

It's important to note that the equation and solution are based on the given data points and the assumption of a linear relationship.

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Answer:

2

Step-by-step explanation:

f(x) = [tex]\frac{5}{2}[/tex] x - 8 ← substitute x = 4 into f(x) , then

f(4) = [tex]\frac{5}{2}[/tex] × 4 - 8 = 5 × 2 - 8 = 10 - 8 = 2

The lateral and the total surface area of the rectangular pyramid are 182 square cm and  278 square cm

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

Length = 6 cm

Width = 8 cm

Height = 6.5 cm

The lateral surface area of the rectangular pyramid is calculated as

LA = 2 *(L + W)H

So, we have

LA = 2 *(6 + 8) * 6.5

Evaluate

LA = 182

The total surface area of the rectangular pyramid is calculated as

TA = 2 * (LW + LH + HW)

So, we have

TA = 2 * (8 * 6 + 8 * 6.5 + 6.5 * 6)

Evalaute

TA = 278

Hence, total surface area of the rectangular pyramid is 278 square cm

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Answer:

-x = 17

Step-by-step explanation:

Given m||n we can write the following equation:

(5x - 23)° + (7x - 1)° = 180° because the these angles are supplementary angles.

(12x - 24)° = 180°

(12x)° = 204°

x = 17

3√2 and 3 are the values of x and y respectively from the figure.

The given diagram is a right triangle with an acute angle of 45 degrees

We need to determine the values of variables x and y.

Applying the trigonometry identity, we will have:

sin 45 = opposite/hypotenuse
sin45 = 3/x

x = 3/sin45

x = 3/(1/√2)
x = 3√2

Similarly:

tan 45 = opposite/adjacent
tan 45 = 3/y

1 = 3/y

y = 3

Hence the values of x and y from the figure is 3√2 and 3 respectively

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The equation of the parabola is (x - 3)² = 20(y - k)

Given data ,

To write the equation of a parabola given its directrix and focus, we can use the standard form for a parabola with a vertical axis of symmetry:

(x - h)² = 4p(y - k)

where (h, k) represents the vertex of the parabola, and the distance between the vertex and the focus is equal to the distance between the vertex and the directrix.

In this case, the directrix is x = -2 and the focus is located at (8, -8). Since the directrix is a vertical line, the parabola has a horizontal axis of symmetry.

And , the vertex lies on the axis of symmetry, which is the line equidistant between the directrix and the focus. In this case, the axis of symmetry is the vertical line x = (8 + (-2)) / 2 = 3.

So, the vertex of the parabola is (3, k), where k is yet to be determined

Next, we need to find the value of p, which represents the distance between the vertex and the focus. Since the focus is at (8, -8), and the vertex is at (3, k), the distance between them is given by

p = 8 - 3 = 5

Now, substituting the values into the standard form equation, we have:

(x - 3)² = 4(5)(y - k)

Simplifying further:

(x - 3)² = 20(y - k)

Hence , the equation of the parabola is (x - 3)^2 = 20(y - k), where k is the y-coordinate of the vertex

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The required measure of point P above the sea level is 6717.22 feet.

Given that,

A is known to be 6,500 feet above sea level; AB = 230 feet.

And, The angle at A looking up at P is 20°.

The angle at B looking up at P is 35°.

Since, These are the equation that contains trigonometric operators such as sin, cos.. etc.. In algebraic operations.

Here,

Let the horizontal distance between A and Q be x and the vertical distance between P and Q be h.

Now,

tan20 = h / x

034x = h

Now,

tan35 = [230 + h]/x

0.7x = 230 + 0.34x

x = 636.89

Now,

h = 0.34 x 636.89

h = 217.2

Now,

The measure of peak P above sea level = 217.2 + 6500 = 6717.22

Thus, the required measure of point P above the sea level is 6717.22 feet.

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The equation of the circle with center (-4, 4) and passing through (-2, 4) is [tex](x + 4)^2 + (y - 4)^2 = 4.[/tex]

We have,

To determine the equation of a circle, we need the center coordinates and the radius. The center coordinates are given as (-4, 4), and we have a point on the circle as (-2, 4).

The distance between the center (-4, 4) and the point on the circle (-2, 4) represents the radius of the circle.

Using the distance formula.

[tex]radius = √[(x_2 - x_1)^2 + (y_2 - y_1)^2]\\= \sqrt{(-2 - (-4))^2 + (4 - 4)^2}\\= \sqrt{2^2 + 0^2}\\= \sqrt{4}\\= 2[/tex]

Now that we have the center coordinates (-4, 4) and the radius 2, we can write the equation of the circle in standard form:

[tex](x - h)^2 + (y - k)^2 = r^2[/tex]

Substituting the values:

[tex](x - (-4))^2 + (y - 4)^2 = 2^2\\(x + 4)^2 + (y - 4)^2 = 4[/tex]

Therefore,

The equation of the circle with center (-4, 4) and passing through (-2, 4) is [tex](x + 4)^2 + (y - 4)^2 = 4.[/tex]

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The wheelchair access ramp must be 216.09 inches long.

To find the length of the wheelchair access ramp, we can use trigonometry.

The tangent function relates the angle of elevation to the ratio of the opposite side (height) to the adjacent side (length of the ramp).

Let's denote the length of the ramp as "x".

The height of the ramp is given as 18 inches.

Using the tangent function:

tan(angle of elevation) = height/length of the ramp

tan(4.8 degrees) = 18/x

To solve for x, we can rearrange the equation:

x = 18 / tan(4.8 degrees)

Using a calculator to evaluate the tangent of 4.8 degrees:

x = 18 / 0.08331

x= 216.09

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The slope of the straight line in the graph that expresses proportional relationship indicates that the lines in the graph that have a slope greater  than 1 but less than 2 are;

The slope of the graph of a straight line is the ratio of the rise to the run on the line.

The slope of a graph with a slope of 1 has an increase in the y-value of 1 for each increase in the x-value of 1

Slope = 1 = Δy/Δx

When the slope is greater than 1, we get;

Δy/Δx > 1, therefore Δy is larger than 1 when Δx is 1.

Similarly, the slope of a graph with a slope of 2 has an increase in the y-value of 2 for each increase in the x-value of 1

Slope = 2 = Δy/Δx

When the slope is less than 1, we get;

Δy/Δx < 2, therefore Δy is less than 2 when Δx is

The lines in the graph that have a slope greater than 1 but less than 2 are therefore the graphs with the coordinates;

Line 3; (0, 0), (4, 6); Slope = 6/4 = 3/2, therefore; 1 < slope = 3/2 < 2

Line 4; (0, 0), (5, 6); Slope = 6/5, therefore; 1 < Slope < 2

The line 5 has a slope of 1, and the line 1, has a slope of 3, line 2 has a slope of 2

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