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.