this is a maximum by taking the second derivative: = −84 + 24x. Next, we plug in x =
1.697 to get approximately = −84 + 24(1.697) = −43.272. Because the value of the
second derivative is negative, according to the second derivative test (see this page), the
volume is a maximum at x = ≈ 1.697 inches.
- 16 meters by 24 meters
First, let’s draw a picture.
If we call the length of the plot y and the width x, the area of the plot is A = xy = 384. The
perimeter is P = 3x + 2y. So, if we want to minimize the length of the fence, we need to
minimize the perimeter of the plot. If we solve the area equation for y, we get y = . Now
we can substitute this for y in the perimeter equation: P = 3x + 2 = 3x + . Now we
take the derivative of P: = 3 − . If we solve this for x, we get x = ±16. A negative
answer doesn’t make any sense in the case, so we use the solution x = 16 meters. Now we can
solve for y: y = = 24 meters.
We can verify that this is a minimum by taking the second derivative: . Next, we
