Math 205 - Calculus I
Homework due November 4
Question 1. The following is question 4.5.1 from our Hass textbook. We
will, as usual, take a step-by-step approach.
What is the smallest perimeter possible for a rectangle whose area is 16 sq.
inches?
(a) Let land wbe the length and width of your rectangle, respectively.
Notice that the area of the rectangle is length times width. Using land
w, how do you write down the condition that the area of the rectangle
is 16?
(b) Write down an expression for perimeter Pin terms of land w.
(c) Use your area condition from (a) to express win terms of l.
(d) Use your expression of Pfrom (b) and expression of wfrom (c) to write
the perimeter Pin terms of only the variable l.
(e) Take a derivative of P(l) and find its critical point.
(f) At this critical point, prove that it is a minimum by showing that the
second derivative P00 is positive at your critical lfrom (e).
(g) Use (a) to compute w. Using your land this w, what is the smallest
possible perimeter?
Question 2. The following is question 4.5.5 from Hass. We will take a
step-by-step approach.
You are planning to make an open-top rectangular box (i.e., a box with no
top) from an 8 in. by 15 in. piece of cardboard by cutting equally-sized
squares from the corners and folding up the resulting flaps up. What are the
dimensions of the box of largest volume you can make this way, and what is
its volume?
(a) Let lbe the length, wthe width, and hthe height. Also, let xbe the
length of the cut that you make when cutting out these squares (as was
done in class). What is l,w, and hin terms of your variable x?
(b) Given that volume is V=lwh, use your results from (a) to write down
volume Vas a function of the variable xonly.
(c) Find a domain for Vby considering what is the minimum and maximum
your cut length xcan be.
(d) Find the critical point(s) of V(x).
(e) Find the end points of your domain.
(f) Use your candidates from (e) and (f) to find the absolute maximum of
V. What xdoes it occur at? What is your length, width, and height?
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