The air resistance R(in pounds) on a racing cyclist is given by the equation R=0.00829s^2 where s is the bicycle's speed (in miles per hour).

a. What is the speed of a racing cyclist who experiences 5 pounds of air resistance?

b. What happens to the air resistance if the cyclist's speed doubles? How would I justify the answer for b algebraically?

.00829s^2 = 5

s^2 = 560.538
s = 23.67

R(2s) = .00829(2s)^2 = .00829*4s^2 = 4R(s)

That is, if s is multiplied by a factor k, R increases by a factor of k^2

To find the speed of a racing cyclist who experiences 5 pounds of air resistance, we can plug in the given value of R into the equation and solve for s.

a. Given R = 5 pounds, we have the equation:
5 = 0.00829s^2

To solve for s, divide both sides of the equation by 0.00829:
5/0.00829 = s^2

Simplifying the left side of the equation, we get:
603.855 throws830 = s^2

Taking the square root of both sides, we find:
s ≈ √603.855 throws830

Therefore, the speed of the racing cyclist who experiences 5 pounds of air resistance is approximately s ≈ 24.6 miles per hour.

b. To determine what happens to the air resistance if the cyclist's speed doubles, let's analyze the equation R = 0.00829s^2.

If the speed doubles, the new speed is 2s. Let's call the new air resistance R'.

Substituting 2s for s in the equation, we have:
R' = 0.00829(2s)^2
= 0.00829(4s^2)
= 0.03316s^2

Comparing R' to the original air resistance R, we see that R' is four times larger than R. Therefore, if the cyclist's speed doubles, the air resistance will increase by a factor of 4.

To find the speed of a racing cyclist who experiences 5 pounds of air resistance, we can use the given equation R = 0.00829s^2.

a. Set R equal to 5 and solve for s:
5 = 0.00829s^2

Divide both sides of the equation by 0.00829:
5 / 0.00829 = s^2

Simplify:
603.144 = s^2

Take the square root of both sides to solve for s:
s = √603.144

Using a calculator, we find that s ≈ 24.57.

Therefore, the speed of a racing cyclist who experiences 5 pounds of air resistance is approximately 24.57 miles per hour.

b. To determine what happens to the air resistance if the cyclist's speed doubles, we need to examine the equation R = 0.00829s^2.

If the cyclist's speed doubles, we substitute 2s for s in the equation and calculate the new value of R.

R = 0.00829(2s)^2
R = 0.00829(4s^2)
R = (0.00829)(4)(s^2)
R = 0.03316s^2

Comparing this to the original equation R = 0.00829s^2, we can conclude that if the cyclist's speed doubles, the air resistance quadruples.

Algebraically, this can be justified by comparing the coefficients in front of the s^2 term. Initially, it is 0.00829, but after doubling the speed, the coefficient becomes 0.03316 (which is 4 times greater).

Therefore, the air resistance would increase by a factor of 4 if the cyclist's speed doubles.