Calculus 1142 Quiz

1. (16 points) According to postal regulations, the girth plus length of parcels sent by
fourth-class mail may not exceed 108 inches. What is the largest possible volume of a
rectangular parcel with two square sides that can be sent by fourth-class mail?

Solution : We have all the quantities described already on the picture. The volume
can be expressed as

We also know that 4x + y = 108 thus we can solve for y and get y = 108 − 4x. The
goal is V -> max. Using the equation for y we get

We also have the bounds on x. For sure x will be positive x > 0. Since 4x + y = 108
we know that x can be at most 27 (for then we have 4x = 108 and y = 0). Compute
the derivative of V (x):

The only critical points will be the points at which V ′(x) = 0 thus we solve x(216 −
12x) = 0 and we get that either x = 0 or x = 18. The number x = 0 is not in our
interval (it is actually one of the end-points). We check the values at the critical points
(x = 18) and at the end-points of the interval. Pay attention to the fact that the
end-points are not included in our interval thus we have to take limits. We get

The maximal volume of the parcel is 11664 in².

2. (4 points) Calculate the following (give exact values!)
(a) (2 points)

Solution : We calculate:

Here we have used the fact that a root is actually a fractional power, a rule of
dividing powers with same bases (subtraction of exponents), and the rule that
describes the power of a power (multiplying the exponents).

(b) (2 points)

Solution: We calculate:

Here we have used the rule of multiplying powers with the same base ( adding
exponents ), and the rule that describes the power of a power ( multiplying the
exponents ).