volume of a block with known are of a square inside

Question

The task is to figure out the volume of a block ABCDEFGH . AB(EH ...) is x and BC(also FG...)is x + 23. The third side is unknown(AE, BF ...) The area of the rectangle BCEH is 4225. sketch

Answer 1

For a rectangular prism, we need to know the length, height, and depth to get the volume. We know the length is $ x $, and the height is $ x+23$. So for volume we have $$ V = x * (x+23) * depth $$ Now, we want to find the depth, which is $EA=FB=GC=HD$. We know the area of the rectangle is 4225. The area of a rectangle has the equation Area = length * height. To make sure we don't get these values confused let's call the $$ Area = length{_{r}} * height{_{r}}$$ The length of the rectangle ($ length{_{r}} $) is $ BC $ which we know to be x+23. The equation for the area is then $ 4225 = (x+23)*height{_{r}} $. Based on the sketch, $height{_{r}}$ is the hypotenuse of the right triangle $ EAB $. Let's rearrange the equation so we get the $height{_{r}}$ in terms of x: $$ \frac{4225}{x+23}=height{_{r}}$$ Remember, we are trying to find the depth to get the volume. The depth is equal to EA, which is the height of the right triangle EAB. Use the Pythagorean Theorem, $ a^2 + b^2 = c^2 $, to solve for the missing side: $$ x^2 + b^2 = (\frac{4225}{x+23})^2 $$ $$ b^2 = (\frac{4225}{x+23})^2 - x^2 $$ $$ b = \sqrt{(\frac{4225}{x+23})^2 - x^2} $$ Plugging in everything in for the volume equation, we get the following: $$ V = x * (x+23) * \sqrt{(\frac{4225}{x+23})^2 - x^2} $$