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