Highwq curves. If a circular curve without any banking has a radius of R feet, the speed limit L in miles per hour is L = 1.5 aqrt(R)
(a) Find the speed limit for a curve having a radius of 400 feet.
(b). If the radius of a curve doubles, what happens to the speed limit?
(c.) A curve with a 40-mile-per-hour speed limit is being designed .What should be its radius?
What is your thinking? I will be happy to critique your thinking.
To obtain the answers to these questions, we will use the given formula: L = 1.5 * sqrt(R), where L represents the speed limit in miles per hour and R represents the radius in feet.
(a) Finding the speed limit for a curve with a radius of 400 feet:
Substituting R = 400 into the formula, we have:
L = 1.5 * sqrt(400)
L = 1.5 * 20
L = 30
Therefore, the speed limit for a curve with a radius of 400 feet is 30 miles per hour.
(b) If the radius of a curve doubles, what happens to the speed limit:
Let's consider the original radius as R and the original speed limit as L.
According to the formula, L = 1.5 * sqrt(R).
If the radius doubles, the new radius would be 2R.
Substituting 2R into the formula, we have:
New speed limit, L' = 1.5 * sqrt(2R)
By substituting sqrt(2R) = sqrt(2) * sqrt(R), we can rewrite the formula as:
L' = 1.5 * sqrt(2) * sqrt(R)
Therefore, when the radius doubles, the new speed limit becomes 1.5 * sqrt(2) times the original speed limit.
(c) Finding the radius for a curve with a 40-mile-per-hour speed limit:
To find the radius for a given speed limit of 40 miles per hour, we need to rearrange the formula and solve for R.
Start with the formula: L = 1.5 * sqrt(R)
Square both sides of the equation to get rid of the square root:
L^2 = (1.5 * sqrt(R))^2
L^2 = (1.5)^2 * (sqrt(R))^2
L^2 = 2.25 * R
To find R, divide both sides of the equation by 2.25:
R = L^2 / 2.25
Substituting L = 40 into the formula, we have:
R = 40^2 / 2.25
R = 1600 / 2.25
R ≈ 711.11 feet
Therefore, for a 40-mile-per-hour speed limit, the radius should be approximately 711.11 feet.