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Section 6.7 Force and Pressure (AI7)

Subsection 6.7.1 Activities

Activity 6.7.2.

Consider a trapezoid-shaped dam that is 60 feet wide at its base and 90 feet wide at its top. Assume the dam is 20 feet tall with water that rises to its top. Water weighs 62.4 pounds per cubic foot and exerts \(P=62.4d\) lbs/ft\(^2\) of pressure at depth \(d\) ft. Consider a rectangular slice of this dam at height \(h_i\) feet and width \(b_i\text{.}\)
A slice at height \(h_i\) of width \(\Delta h\text{,}\) with base \(b_i\) of a damn with base 60 ft, top 90 ft, 20 ft tall.
Figure 159. A slice at height \(h_i\) of width \(\Delta h\text{.}\)
(a)
At a height of \(h_i\) feet, what is the base of the rectangle \(b_i\text{?}\)
(b)
What is the area of a rectangle with base \(b_i\) feet and height \(\Delta h\) feet?
(c)
Using a depth of \(20-h_i\) feet, how much pressure is exerted on this rectangle?
(d)
Using the pressure found in (c), the area in (b), and Fact 6.7.1, how much force is exerted on this rectangle?

Activity 6.7.3.

Recall the computations done in Activity 6.7.2.
(a)
Find a Riemann sum which estimates the total force exerted on the dam, using slices at heights \(h_i\) m, of width \(\Delta h\) m.
(b)
Use (a) to find an integral expression which computes the amount of force exerted on this dam.
(c)
Evaluate the integral found in (b).

Subsection 6.7.2 Videos

Figure 160. Video: Set up integrals to solve problems involving work, force, and/or pressure