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Calculus 1142 Quiz
1. (16 points) According to postal regulations, the girth plus length of
parcels sent by
fourthclass 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 fourthclass 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 endpoints). We check the values at the
critical points
(x = 18) and at the endpoints of the interval. Pay attention to the fact that
the
endpoints 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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