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?
a. R = 0.00829s^2 = 5Lbs
s^2 = 603.14
s = 24.6 mi/h.
b. R = 0.008299*(2s)^2 = 4*0.00829s^2
The resistance increases by a factor of 4.
a. To find the speed of a racing cyclist who experiences 5 pounds of air resistance, we can substitute R = 5 into the equation R = 0.00829s^2 and solve for s:
5 = 0.00829s^2
Divide both sides by 0.00829:
s^2 = 5 / 0.00829
s^2 ≈ 603.8612
Take the square root of both sides:
s ≈ √603.8612
s ≈ 24.58 mph
Therefore, the speed of the racing cyclist who experiences 5 pounds of air resistance is approximately 24.58 mph.
b. If the cyclist's speed doubles, we need to determine what happens to the air resistance. Let's consider the new speed as 2s (since it doubles).
Using the equation R = 0.00829s^2, we'll substitute s with 2s:
R = 0.00829(2s)^2
R = 0.00829(4s^2)
R = 0.03316s^2
As we can see, when the speed doubles, the air resistance is multiplied by 0.03316. In other words, the air resistance increases by approximately 3.316 times (or around 231.6%).
So, if the cyclist's speed doubles, the air resistance increases significantly.
Now, it's important to note that this is an approximation based on the given equation. Actual air resistance may vary due to factors like wind, aerodynamics, terrain, etc.
a. To find the speed of a racing cyclist who experiences 5 pounds of air resistance, we can set up the equation R = 5 and solve for s.
Given:
R = 0.00829s^2 (equation for air resistance)
R = 5 (given air resistance value)
Substituting the given values into the equation, we get:
0.00829s^2 = 5
Now, divide both sides of the equation by 0.00829:
s^2 = 5 / 0.00829
Simplifying the right-hand side, we get:
s^2 ≈ 604.573
Taking the square root of both sides, we find:
s ≈ √604.573
Therefore, the speed of the racing cyclist who experiences 5 pounds of air resistance is approximately equal to the square root of 604.573, in miles per hour.
b. To determine how the air resistance changes when the cyclist’s speed doubles, we need to evaluate the new air resistance using the equation R = 0.00829s^2, where s represents the new speed.
Given:
R₁ = 0.00829s₁^2 (equation for initial air resistance)
R₂ = 0.00829s₂^2 (equation for new air resistance after speed doubles)
Since we are doubling the speed, we can express it as s₂ = 2s₁.
Substituting this value into the equation for the new air resistance, we get:
R₂ = 0.00829(2s₁)^2
Expanding and simplifying this expression, we have:
R₂ = 0.00829 * 4s₁^2
= 0.03316s₁^2
By comparing this equation to the equation for the initial air resistance, we see that the new air resistance (R₂) is four times greater than the initial air resistance (R₁) when the speed doubles.
Algebraically, we justified the answer by representing the new speed as twice the initial speed and substituted it into the equation for air resistance. By simplifying the expression, we determined that the new air resistance is four times greater.
a. To find the speed of a racing cyclist who experiences 5 pounds of air resistance, we can set up the equation:
R = 0.00829s^2
Given that R = 5 pounds, we can substitute this into the equation:
5 = 0.00829s^2
Now, we can solve for s. Rearrange the equation:
0.00829s^2 = 5
Divide both sides of the equation by 0.00829:
s^2 = 5 / 0.00829
s^2 = 604.8209
Taking the square root of both sides:
s ≈ √(604.8209)
s ≈ 24.61
Therefore, the speed of the racing cyclist who experiences 5 pounds of air resistance is approximately 24.61 miles per hour.
b. To determine what happens to the air resistance if the cyclist's speed doubles, we need to look at the equation R = 0.00829s^2.
If the speed doubles, we can represent it as 2s. Now let's substitute this value into the equation to see the effect on R:
R = 0.00829(2s)^2
R = 0.00829(4s^2)
R = 0.03316s^2
Comparing this result to the original equation, we can see that when the speed doubles, the air resistance increases by a factor of 4 (0.00829 × 4 = 0.03316). Therefore, the air resistance quadruples.
Algebraically, we justified the answer by substituting the doubled speed (2s) into the equation and simplifying the expression to demonstrate the impact on air resistance.