Atriangular plate with height 6 ft and a base of 7 ft is submerged vertically in water so that the top is 2 ft below the surface. express the hydrostatic force against one side of the plate as an integral and evaluate it. (recall that the weight density of water is 62.5 lb/ft3.)

Explanation:

For this problem to be easier to calculate, we can represent the triangle as a right triangle whose right angle is located at the origin of a coordinate system. (See picture attached).

With this disposition of the triangle, we can start finding our integral. The hydrostatic force can be set as an integral with the following shape:

γhxdy

we know that γ=62.5 lb/

from the drawing, we can determine the height (or depth under the water) of each differential area is given by:

h=8-y

x can be found by getting the equation of the line, which we'll get by finding the slope of the line and using one of the points to complete the equation:

when substituting the x and y-values given on the graph, we get that the slope is:

once we got this slope, we can substitute it in the point-slope form of the equation:

which yields:

which simplifies to:

we can now solve this equation for x, so we get that:

with this last equation, we can substitute everything into our integral, so it will now look like this:

Now that it's all written in terms of y we can now simplify it, so we get:

we can now proceed and evaluate it.

When using the power rule on each of the terms, we get the integral to be:

By using the fundamental theorem of calculus we get:

When solving we get:

you cant move and you feel like someones pushing you down onto your bed it can feel like you want to get up but you just can't it can be extremely scary