# 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 in^{2}.

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 ).

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