Application Problem for MATH 1401 [Calculus I]
Dear Mike:
We hear that you need to design some heavy-duty rain gutters for the new addition to the
Science Building. We have designed some models that you should consider. According
to the information given in the Calculus course, we know that the capacity for a gutter
depends on its cross-sectional area. The bottom of the cross-section is a fixed length and
we were told that you want this fixed length to be 20 inches.
[For each model below, explain the geometry of the cross-section. Take the derivative
and set it equal to zero. Show that the critical number is a maximum.]
MODEL I: Semicircle
[Sketch Figure 1 which is a semicircle of arc length 20 inches. The top of the figure
should be a diameter and you should use a dashed line for this. In your report, calculate
the radius and area of the semicircular region. No optimization here, just algebra.]
MODEL II: Isosceles Triangle
[Sketch Figure 2 which has the base of the triangle on top (dashed). The angle between
the two ten-inch sides is theta. Now redraw the triangle in Figure 3 which has one of the
ten-inch sides horizontal and theta is in the bottom left-hand corner of the triangle. From
this figure, you can write a formula for area = (1/2)*(base)*(height) and then determine
the best value of theta which gives you the maximum area.]
MODEL III: Rectangle
[Sketch Figure 4 which has two vertical sides of length x and a bottom base of length
(20 – 2x). The upper base should be dashed. Find the value of x which gives you the
maximum area.
MODEL IV: Trapezoid
[Sketch Figure 5 as shown in class. The two sides have length 5 and the lower base has
length 10. Find the value of theta (shown in class) which gives the maximum area. In
other words, can you improve on the rectangle from the previous model?]
[What is your conclusion about designing the gutters?]
[Sign it!]