the Ohio turnpike has a maximum uphill slope of 3 degrees. how long must a straight uphill segment of the road be in order to allow a vertical rise of 450 feet

450=distance*sin3deg

To find the length of a straight uphill segment of the road required to allow a vertical rise of 450 feet, we can use trigonometry.

We know that the maximum uphill slope of the Ohio Turnpike is 3 degrees. The maximum uphill slope can be represented as the ratio of the vertical rise to the horizontal distance.

Let's assume the length of the uphill segment is "L" (in feet).

Now, we can set up the equation using the trigonometric relationship:

tan(3 degrees) = vertical rise / horizontal distance

Plugging in the values:

tan(3 degrees) = 450 / L

To solve for L, we can rearrange the equation:

L = 450 / tan(3 degrees)

Using a scientific calculator, tan(3 degrees) is approximately 0.052406.

L = 450 / 0.052406

Simplifying:

L ≈ 8,589 feet

Therefore, the length of a straight uphill segment of the road should be approximately 8,589 feet to allow a vertical rise of 450 feet, given that the maximum uphill slope is 3 degrees.

To calculate the length of a straight uphill segment of the road, we can use trigonometry. The relationship between the angle, the vertical rise (height), and the length of the road is given by the tangent function:

tan(angle) = height / length

Rearranging the formula, we have:

length = height / tan(angle)

In this case, we are given that the uphill slope has a maximum angle of 3 degrees and a vertical rise of 450 feet.

First, we need to convert the angle from degrees to radians because trigonometric functions typically work with radians.

angle = 3 degrees * (π / 180) radians/degree ≈ 0.052 radians

Now we can calculate the length of the road:

length = 450 feet / tan(0.052 radians)

To find the answer, we need to use a calculator or a tool with a tangent function. Plugging the numbers into a calculator, we get:

length ≈ 450 feet / 0.052 ≈ 8654 feet

Therefore, in order to allow a vertical rise of 450 feet, the straight uphill segment of the road must be approximately 8,654 feet long.