Math 212
Name: __________________________________
Homework 3
1. A long sheet of metal with a width of 12 inches is folded into the shape of a trough. The following
picture shows a cross-section of the trough:
The two side pieces each have width B, and are inclined at an angle of ). The amount of water that
the trough can carry is determined by the area E of the resulting trapezoidal cross section.
(a) Find a formula for the area E in terms of B and ).
(b) Make a contour plot for the function EaBß )b on the following grid. Include the contours for
! in# , & in# , "! in# , "& in# , and #! in# . (Feel free to use a calculator or computer for this part.)
PRACTICE SPACE
FINAL ANSWER
А2
Θ HradiansL
Θ HradiansL
А2
А4
0
А4
0
0
1
2
3
x HinchesL
4
5
6
0
1
2
3
x HinchesL
4
5
6
(c) Find formulas for the partial derivatives
`E
`E
and
in terms of B and ).
`B
`)
(d) Using your formulas from part (c), find the values of B and ) that maximize the cross-sectional
area E. (Hint: You may need to use the identity sin# ) œ "  cos# ) to solve the equations.)