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.)
2. The goal of this problem is to estimate the value of the integral
(( ÈB$  C$ .E
V
where V is the rectangle eaBß Cb ± ! Ÿ B Ÿ "Þ&ß ! Ÿ C Ÿ "Þ&f.
(a) Use nine subrectangles and the Midpoint Rule to estimate the value of this integral. Record the
height that you are using for each box on the table below:
1.5
1.0
0.5
0.0
0.0
0.5
1.0
1.5
Note: If you do this part correctly, your answer should be approximately #Þ&(!).
(b) Use a spreadsheet such as Microsoft Excel to estimate this integral using 900 rectangles (30 in
each direction) and the Midpoint Rule. Write your final answer below, and attach a printout of
your spreadsheet to your homework assignment.
3. Cottage Lake is a small lake in King County, Washington. The following figure shows a
bathymetric chart for the lake, i.e. a contour plot of the depth of the water. Depths are given in
feet, and each square has a side length of 500 feet:
Using this contour plot, estimate the total volume of water in Cottage Lake as accurately as you can.
(Show your work, and express your final answer in cubic feet. Your answer must be correct to
within 20% to receive full credit.)