A company introduces a new product for which the number of units sold S is given by the equation below, where t is the time in months.
s(t)=155(7-9/(2+t))
a) Find the average rate of change of s(t) during the first year.
Which my answer was 1395/28
b) During what month of the first year does s'(t) equal the average rate of change?
I worked out the problem and got t to equal around 3.29, rounding that this would be the month of March. I entered that answer but the website didn't take that as correct. What could I have done wrong??
Got the same answer, 3.29 which is around march, but that answer was not correct according to the website
Hmm, it seems like there might be a mistake in your calculations. Let's solve the problem step by step to see where the error might have occurred.
a) To find the average rate of change of s(t) during the first year, we need to find the difference between s(12) and s(0), and divide it by the number of months (12) in the first year:
s(12) = 155(7 - 9/(2+12)) = 155(7 - 9/14) = 155(7 - 9/14) = 155 * (98/14 - 9/14) = 155 * (89/14) = 1395/2.
s(0) = 155(7 - 9/(2+0)) = 155(7 - 9/2) = 155(7 - 4.5) = 155 * 2.5 = 387.5.
Therefore, the average rate of change of s(t) during the first year is:
(1395/2 - 387.5) / 12 = 1395/28, which is the correct answer you provided.
b) To find when s'(t) equals the average rate of change, we need to find the value of t that satisfies the equation:
s'(t) = 1395/28.
Now, let's differentiate the equation for s(t) with respect to t:
s(t) = 155(7 - 9/(2+t))
s'(t) = 155(-9/(2+t)^2)
Now we can equate s'(t) to the average rate of change:
155(-9/(2+t)^2) = 1395/28.
To simplify the equation, we can cancel out common factors:
-9/(2+t)^2 = 9/4.
Now, let's cross-multiply and solve for t:
-9 * 4 = 9 * (2+t)^2.
-36 = 9*(2+t)^2.
(2+t)^2 = -36/9.
(2+t)^2 = -4.
Uh-oh! Looks like we have a problem here. The equation (2+t)^2 = -4 has no real solutions since the square of a number can never be negative. This means that s'(t) will never equal the average rate of change.
So, it seems that there might be an error in the problem statement or in your calculations for part b. Double-check your work or consult your instructor for further clarification.
To find the average rate of change of s(t) during the first year, you need to evaluate s(t) at the endpoints of the year (t=0 and t=12) and then divide the change in s(t) by the change in t.
a) Average rate of change = (s(12) - s(0)) / (12 - 0)
Plug in t=12:
s(12) = 155(7-9/(2+12)) = 155(7-9/14) = 155(7-9/14) = 155(7-9/14) = 155(91/14 - 9/14) = 155 * 82/14 = 155 * 41/7 = 1435
Plug in t=0:
s(0) = 155(7-9/(2+0)) = 155(7-9/2) = 155(7-4.5) = 155 * 2.5 = 387.5
Average rate of change = (1435 - 387.5) / (12-0) = 1047.5 / 12 = 87.29
So the average rate of change during the first year is approximately 87.29 units per month.
b) To find the month in the first year when s'(t) equals the average rate of change, you need to find the value of t when the derivative of s(t) equals the average rate of change.
s'(t) = -155 * [9/(2+t)^2]
Setting s'(t) equal to the average rate of change:
-155 * [9/(2+t)^2] = 87.29
Multiply both sides by (-1) and reciprocal:
[9/(2+t)^2] = -87.29/155
Cross-multiply:
9 = -87.29 * (2+t)^2 / 155
Multiply both sides by 155:
9 * 155 = -87.29 * (2+t)^2
Divide both sides by -87.29:
(9 * 155) / (-87.29) = (2+t)^2
Take the square root of both sides:
√[ (9 * 155) / (-87.29) ] = √(2+t)^2
Simplify the square root:
±√[ (9 * 155) / (-87.29) ] = 2 + t
Since t represents time, it cannot be negative. Therefore, we can ignore the negative sign. So, we have:
√[ (9 * 155) / (-87.29) ] = 2 + t
Now, square both sides:
(9 * 155) / (-87.29) = (2 + t)^2
Simplify:
(9 * 155) / (-87.29) = (2 + t)^2
To solve for t, you need to solve this equation. However, the value of t will depend on the specific numbers used in the equation. Please recheck the values you used for 9, 155, and -87.29 and ensure your calculations are correct.
To find the average rate of change of s(t) during the first year, you need to calculate s(12) - s(0) and divide it by 12, as the time period is one year.
a) Average rate of change = (s(12) - s(0)) / 12
To find s(12), substitute t = 12 into the equation s(t):
s(12) = 155(7 - 9/(2+12))
Calculating this expression gives you s(12).
Similarly, calculate s(0) by substituting t = 0 into the equation s(t):
s(0) = 155(7 - 9/(2+0))
Calculating this expression gives you s(0).
Once you know both s(12) and s(0), calculate the average rate of change by dividing s(12) - s(0) by 12.
b) To find the month during the first year when s'(t) equals the average rate of change, you need to find t such that s'(t) = average rate of change.
Calculate s'(t) by taking the derivative of s(t) with respect to t. Once you have the derivative, set it equal to the average rate of change you found in part (a). Solve the equation to find t.
If you rounded the answer for t to be around 3.29, then the month would be March (as the question asks for the month).
It's possible that you made a calculation error while finding s'(t) or solving the equation. Check your steps and calculations again to make sure you didn't make any mistakes.
Part (a)
average rate of change is the difference between the end of last month to the beginning of the first month, divided by the number of months, so
Average=(s(12)-s(0))/12
Part (b)
Calculate the derivative of the function s(t), with respect to t:
d s(t) / dt
=d(1085-1395/(2+t))/dt
= 0 + 1395/(2+t)²
=1395/(2+t)²
by applying the power and chain rules.
Then equate the rate of change function (derivative) with the expression for the average over 12 months, which gives
1395/(2+t)²=1395/28
To solve for t, cross multiply and solve the quadratic for t, which should give two solutions, one of which is negative and to be rejected. The other should be between 0 to 12.
Post if you wish an answer check.