The fuel efficiency for a certain midsize car is given by
E(v) = −0.017v^ + 1.462v + 3.5 where E(v)is the fuel efficiency in miles per gallon for a car traveling v miles per hour.
(a) What speed will yield the maximum fuel efficiency? Round to the nearest mile per hour. _______mph
(b) What is the maximum fuel efficiency for this car? Round to the nearest mile per gallon. _______mpg
You have a typo, there is no exponent on the first term
anyway ...
max when E ' (v) = 0
E ' (v) = .....
then set that equal to zero and solve for v
for b) sub in your v from a) into the original equation
This is how the problem reads.
A = 43mph but I'm not sure how to solve B
To find the speed that will yield the maximum fuel efficiency for the given equation, we need to find the maximum value of the function E(v). This can be done by taking the derivative of E(v) with respect to v and setting it equal to zero.
(a) To find the speed, v, that yields the maximum fuel efficiency, we need to find the value of v where the derivative of E(v) equals zero.
Step 1: Take the derivative of E(v) with respect to v:
E'(v) = -0.034v + 1.462
Step 2: Set E'(v) equal to zero and solve for v:
-0.034v + 1.462 = 0
-0.034v = -1.462
v = -1.462 / -0.034
v ≈ 43 (rounded to the nearest mph)
Therefore, the speed that will yield the maximum fuel efficiency is approximately 43 mph.
(b) To find the maximum fuel efficiency, we need to substitute the value of v we found in part (a) into the original equation E(v).
Step 1: Substitute v = 43 into E(v):
E(43) = -0.017(43)^2 + 1.462(43) + 3.5
E(43) = -31.927 + 63.066 + 3.5
E(43) ≈ 34.64 (rounded to the nearest mpg)
Therefore, the maximum fuel efficiency for this car is approximately 34.64 mpg.