The isosceles triangular plate shown here is submerged vertically

22

ft below the surface of a freshwater lake.

**a.** Find the fluid force against one face of the plate.

**b.** What would be the fluid force on one side of the plate if the water were seawater instead of​ freshwater?

Surface Level6 ft6 ft2 ft

​(The triangle is not shown to​ scale.)

Question content area bottom

Part 1

**a.** Establish a coordinate system to work in by placing the origin at the​ plate's bottom vertex and running the​ y-axis upward along the​ plate's axis of symmetry. Assuming the triangle is submerged in​ freshwater, set up the integral that represents the fluid force against one face of the plate.

Integral from 0 to nothing left parenthesis nothing right parenthesis dy

To solve this fluid force problem, let's break it down step by step. The force \( F \) exerted by a fluid on a submerged surface is given by the following formula:

\[
F = \int_{a}^{b} p(y) \, w(y) \, dy
\]

Where:
- \( p(y) \) is the pressure at depth \( y \).
- \( w(y) \) is the width of the plate at depth \( y \).
- \( y \) is the depth below the surface of the fluid.

### Step 1: Understanding the Geometry of the Problem
The plate is an isosceles triangle submerged 22 ft below the surface of the lake, with a height of 6 ft and a base of 2 ft.

1. **Coordinate System**: Place the origin at the bottom vertex of the triangular plate. The height of the plate extends from \( y = 0 \) (bottom of the plate) to \( y = 6 \) ft (top of the plate).
2. **Pressure Function**: The pressure at depth \( y \) is due to the weight of the water above the point, and is given by:

\[
p(y) = \rho g (22 + y)
\]
Where:
- \( \rho \) is the density of freshwater, approximately \( 62.4 \, \text{lb/ft}^3 \).
- \( g \) is the acceleration due to gravity (which is included in \( \rho \)).
- \( 22 + y \) is the depth below the surface (since the plate is submerged 22 ft below).

### Step 2: Finding the Width \( w(y) \)
The triangle is isosceles with a base of 2 ft at the top, tapering down to a width of 0 ft at the bottom. The relationship between the width \( w(y) \) and the height \( y \) is linear.

- The slope of the triangle's

The isosceles triangular plate shown here is submerged vertically 22 ft below the surface of a freshwater lake. 
**a.** Find the fluid force against one face of the plate. 
**b.** What would be the fluid force on one side of the plate if the water were seawater instead of freshwater?
Surface Level6 ft6 ft2 ft
(The triangle is not shown to scale.)

**Part 1**
**a.** Establish a coordinate system to work in by placing the origin at the plate's bottom vertex and running the y-axis upward along the plate's axis of symmetry. Assuming the triangle is submerged in freshwater, set up the integral that represents the fluid force against one face of the plate.

Learn how to calculate the fluid force exerted on a submerged isosceles triangular plate 22 ft below the surface of a freshwater lake. This problem uses integral calculus to set up and solve for the fluid force, covering key concepts like hydrostatic pressure and properties of submerged surfaces. Explore the step-by-step solution and find out how the force changes when the fluid is seawater.

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