The following polynomial function models the speed of a swimmer doing the breast stroke during one complete stroke, where t is the number of seconds since the start of the stroke.
S= -241t^7 + 1062t^6 -1871t^5 +1647t^4 - 737t^3 + 144t^2 -2.432t
A) At what time is the swimmer going the fastest?
B) How fast is the swimmer going?
C) How long does it take the swimmer to complete one full stroke?
did you ever get this answer?
To find the answers to these questions, we need to differentiate the function S(t) with respect to time (t) and then solve for the corresponding values.
A) To find the time when the swimmer is going the fastest, we need to find the value of t when the derivative of S(t) is equal to zero.
So, let's differentiate S(t) with respect to t:
dS(t)/dt = d/dt(-241t^7 + 1062t^6 - 1871t^5 + 1647t^4 - 737t^3 + 144t^2 - 2.432t)
To do this, we can use the power rule of differentiation. The power rule states that if the function is of the form f(t) = at^n, then its derivative is given by f'(t) = an*t^(n-1).
Using the power rule, we differentiate each term:
dS(t)/dt = -241*7t^(7-1) + 1062*6t^(6-1) - 1871*5t^(5-1) + 1647*4t^(4-1) - 737*3t^(3-1) + 144*2t^(2-1) - 2.432(1)
Simplifying the expression, we get:
dS(t)/dt = -1687t^6 + 6372t^5 - 9355t^4 + 6588t^3 - 2211t^2 + 144t - 2.432
Now we need to solve for t when dS(t)/dt = 0:
-1687t^6 + 6372t^5 - 9355t^4 + 6588t^3 - 2211t^2 + 144t - 2.432 = 0
To solve this equation, we can use numerical methods or an algebraic calculator to find the values of t. The solution(s) will give us the time(s) when the swimmer is going the fastest.
B) To find how fast the swimmer is going when they are at their fastest, we need to substitute the values of t we found in part A back into the original function S(t). This will give us the corresponding speed.
C) To find how long it takes for the swimmer to complete one full stroke, we need to find the time it takes for the function S(t) to return to the same value it started with at t = 0. In other words, we need to find the time when S(0) = S(t).
To solve for this, we can set S(0) equal to S(t) and solve for t:
-241*(0)^7 + 1062*(0)^6 - 1871*(0)^5 + 1647*(0)^4 - 737*(0)^3 + 144*(0)^2 - 2.432*(0) = -241t^7 + 1062t^6 - 1871t^5 + 1647t^4 - 737t^3 + 144t^2 - 2.432t
Simplifying the equation, we get:
0 = -241t^7 + 1062t^6 - 1871t^5 + 1647t^4 - 737t^3 + 144t^2 - 2.432t
Similar to part A, we can use numerical methods or an algebraic calculator to find the value(s) of t that solve this equation. The solution(s) will give us the time it takes for the swimmer to complete one full stroke.