Solved on Feb 15, 2024

Calculate the percent grade and angle of elevation for a highway with a vertical rise of $140$ feet over $2000$ feet of horizontal distance.

STEP 1

Assumptions
1. The vertical rise of the highway is 140 feet.
2. The horizontal distance of the highway is 2000 feet.
3. Percent grade is calculated as the ratio of vertical rise to horizontal distance, expressed as a percentage.
4. The angle of elevation is the angle formed by the horizontal distance and the line of the slope (rise).

STEP 2

First, we need to calculate the percent grade. The percent grade is given by the formula:
$\text{Percent grade} = \left(\frac{\text{Vertical rise}}{\text{Horizontal distance}}\right) \times 100\%$

STEP 3

Now, plug in the given values for the vertical rise and horizontal distance to calculate the percent grade.
$\text{Percent grade} = \left(\frac{140\, \text{feet}}{2000\, \text{feet}}\right) \times 100\%$

STEP 4

Perform the division to find the ratio.
$\text{Percent grade} = \left(\frac{140}{2000}\right) \times 100\%$
$\text{Percent grade} = 0.07 \times 100\%$

STEP 5

Calculate the percent grade.
$\text{Percent grade} = 0.07 \times 100\% = 7\%$

STEP 6

Now, we need to find the angle of elevation. The angle of elevation can be found using the tangent function, since the tangent of an angle in a right triangle is the ratio of the opposite side (vertical rise) to the adjacent side (horizontal distance).
$\tan(\text{Angle of elevation}) = \frac{\text{Vertical rise}}{\text{Horizontal distance}}$

STEP 7

Plug in the given values for the vertical rise and horizontal distance to find the tangent of the angle of elevation.
$\tan(\text{Angle of elevation}) = \frac{140\, \text{feet}}{2000\, \text{feet}}$

STEP 8

Perform the division to find the tangent of the angle of elevation.
$\tan(\text{Angle of elevation}) = \frac{140}{2000}$
$\tan(\text{Angle of elevation}) = 0.07$

STEP 9

To find the angle of elevation, we need to take the arctangent (inverse tangent) of the tangent value we calculated.
$\text{Angle of elevation} = \arctan(0.07)$

STEP 10

Use a calculator or a trigonometric table to find the arctangent of 0.07 to get the angle of elevation in degrees.
$\text{Angle of elevation} = \arctan(0.07) \approx 4.01^\circ$

STEP 11

Round the angle of elevation to the nearest whole number if necessary.
$\text{Angle of elevation} \approx 4^\circ$
The percent grade of the highway is 7% and the angle of elevation is approximately 4 degrees.

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Calculate the percent grade and angle of elevation for a highway with a vertical rise of $140$ feet over $2000$ feet of horizontal distance.

STEP 1

STEP 2

First, we need to calculate the percent grade. The percent grade is given by the formula:
$\text{Percent grade} = \left(\frac{\text{Vertical rise}}{\text{Horizontal distance}}\right) \times 100\%$

STEP 3

Now, plug in the given values for the vertical rise and horizontal distance to calculate the percent grade.
$\text{Percent grade} = \left(\frac{140\, \text{feet}}{2000\, \text{feet}}\right) \times 100\%$

STEP 4

Perform the division to find the ratio.
$\text{Percent grade} = \left(\frac{140}{2000}\right) \times 100\%$
$\text{Percent grade} = 0.07 \times 100\%$

STEP 5

Calculate the percent grade.
$\text{Percent grade} = 0.07 \times 100\% = 7\%$

STEP 6

STEP 7

Plug in the given values for the vertical rise and horizontal distance to find the tangent of the angle of elevation.
$\tan(\text{Angle of elevation}) = \frac{140\, \text{feet}}{2000\, \text{feet}}$

STEP 8

Perform the division to find the tangent of the angle of elevation.
$\tan(\text{Angle of elevation}) = \frac{140}{2000}$
$\tan(\text{Angle of elevation}) = 0.07$

STEP 9

To find the angle of elevation, we need to take the arctangent (inverse tangent) of the tangent value we calculated.
$\text{Angle of elevation} = \arctan(0.07)$

STEP 10

Use a calculator or a trigonometric table to find the arctangent of 0.07 to get the angle of elevation in degrees.
$\text{Angle of elevation} = \arctan(0.07) \approx 4.01^\circ$

STEP 11

Round the angle of elevation to the nearest whole number if necessary.
$\text{Angle of elevation} \approx 4^\circ$
The percent grade of the highway is 7% and the angle of elevation is approximately 4 degrees.

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Calculate the percent grade and angle of elevation for a highway with a vertical rise of 140140140 feet over 200020002000 feet of horizontal distance.

STEP 1

STEP 2

STEP 3

Now, plug in the given values for the vertical rise and horizontal distance to calculate the percent grade.Percent grade=(140 feet2000 feet)×100%\text{Percent grade} = \left(\frac{140\, \text{feet}}{2000\, \text{feet}}\right) \times 100\%Percent grade=(2000feet140feet​)×100%

STEP 4

Perform the division to find the ratio.Percent grade=(1402000)×100%\text{Percent grade} = \left(\frac{140}{2000}\right) \times 100\%Percent grade=(2000140​)×100%Percent grade=0.07×100%\text{Percent grade} = 0.07 \times 100\%Percent grade=0.07×100%

STEP 5

Calculate the percent grade.Percent grade=0.07×100%=7%\text{Percent grade} = 0.07 \times 100\% = 7\%Percent grade=0.07×100%=7%

STEP 6

STEP 7

Plug in the given values for the vertical rise and horizontal distance to find the tangent of the angle of elevation.tan⁡(Angle of elevation)=140 feet2000 feet\tan(\text{Angle of elevation}) = \frac{140\, \text{feet}}{2000\, \text{feet}}tan(Angle of elevation)=2000feet140feet​

STEP 8

Perform the division to find the tangent of the angle of elevation.tan⁡(Angle of elevation)=1402000\tan(\text{Angle of elevation}) = \frac{140}{2000}tan(Angle of elevation)=2000140​tan⁡(Angle of elevation)=0.07\tan(\text{Angle of elevation}) = 0.07tan(Angle of elevation)=0.07

STEP 9

To find the angle of elevation, we need to take the arctangent (inverse tangent) of the tangent value we calculated.Angle of elevation=arctan⁡(0.07)\text{Angle of elevation} = \arctan(0.07)Angle of elevation=arctan(0.07)

STEP 10

Use a calculator or a trigonometric table to find the arctangent of 0.07 to get the angle of elevation in degrees.Angle of elevation=arctan⁡(0.07)≈4.01∘\text{Angle of elevation} = \arctan(0.07) \approx 4.01^\circAngle of elevation=arctan(0.07)≈4.01∘

STEP 11

Round the angle of elevation to the nearest whole number if necessary.Angle of elevation≈4∘\text{Angle of elevation} \approx 4^\circAngle of elevation≈4∘The percent grade of the highway is 7% and the angle of elevation is approximately 4 degrees.

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Calculate the percent grade and angle of elevation for a highway with a vertical rise of $140$ feet over $2000$ feet of horizontal distance.

Now, plug in the given values for the vertical rise and horizontal distance to calculate the percent grade.
$\text{Percent grade} = \left(\frac{140\, \text{feet}}{2000\, \text{feet}}\right) \times 100\%$

Perform the division to find the ratio.
$\text{Percent grade} = \left(\frac{140}{2000}\right) \times 100\%$
$\text{Percent grade} = 0.07 \times 100\%$

Calculate the percent grade.
$\text{Percent grade} = 0.07 \times 100\% = 7\%$

Plug in the given values for the vertical rise and horizontal distance to find the tangent of the angle of elevation.
$\tan(\text{Angle of elevation}) = \frac{140\, \text{feet}}{2000\, \text{feet}}$

Perform the division to find the tangent of the angle of elevation.
$\tan(\text{Angle of elevation}) = \frac{140}{2000}$
$\tan(\text{Angle of elevation}) = 0.07$

To find the angle of elevation, we need to take the arctangent (inverse tangent) of the tangent value we calculated.
$\text{Angle of elevation} = \arctan(0.07)$

Use a calculator or a trigonometric table to find the arctangent of 0.07 to get the angle of elevation in degrees.
$\text{Angle of elevation} = \arctan(0.07) \approx 4.01^\circ$

Round the angle of elevation to the nearest whole number if necessary.
$\text{Angle of elevation} \approx 4^\circ$
The percent grade of the highway is 7% and the angle of elevation is approximately 4 degrees.