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

### Given Information:

- The triangular plate is submerged 22 ft below the surface of a freshwater lake.
- The triangle is isosceles with a base of 6 ft and a height of 6 ft.
- Freshwater has a weight density of \( \rho = 62.4 \, \text{lb/ft}^3 \).
- We are asked to find the fluid force against one face of the plate using integrals.

### Step 1: Coordinate System and Geometry Setup

To simplify the problem, we place the origin of the coordinate system at the **bottom vertex** of the triangle. The \(y\)-axis runs upward along the plate's axis of symmetry. 

- The height of the plate (from the bottom vertex to the base) is 6 ft. 
- The base of the plate is 6 ft wide, which means the half-width of the plate at any height \(y\) is 3 ft (symmetrical about the \(y\)-axis).

The equation for the half-width of the plate at height \(y\) can be derived using the geometry of the triangle. Since the plate is isosceles:
\[
\frac{x(y)}{6} = \frac{y}{6} \quad \text{(linear relationship between width and height)} 
\]
This simplifies to:
\[
x(y) = y
\]
So, the total width of the plate at height \(y\) is:
\[
\text{Width at } y = 2x(y) = 2y
\]

### Step 2: Depth of the Plate
The plate is submerged 

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 against a submerged isosceles triangular plate 22 ft below the surface of a freshwater lake using integrals. The solution involves fluid mechanics and hydrostatic pressure formulas, suitable for college-level calculus or physics students.

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