Video: Writing an Algebraic Expression for the Volume of a Rectangular Prism given the Area of Its Base and the Height

Bethani Gasparine

The area of the floor of a silo is (2π‘₯ + 9)Β² ftΒ². The height of the silo is (10π‘₯ + 10) ft. Write an expression for the volume of grain that the silo can hold by expanding the square and multiplying by the height.

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Video Transcript

The area of the floor of a silo is two π‘₯ plus nine squared square feet. The height of the silo is 10π‘₯ plus 10 feet. Write an expression for the volume of grain that the silo can hold by expanding the square and multiplying by the height.

So we are to find the volume of the silo. And volume has a formula the area of the base times the height. And we know that the area of the floor of the silo, which would be the base, is equal to two π‘₯ plus nine squared square feet. And then the height we know to be 10π‘₯ plus 10 feet.

So it told us to find the volume of grain that the silo can hold by expanding the square β€” so we will be expanding this square β€” and then multiplying by the height. So we will do exactly that.

Now we won’t keep writing feet squared and feet. If we know that our answer should have feet squared times feet, that would be feet cubed. And that makes sense because a volume should be cubic feet. So we will add that to the end.

So we will first begin to expand by writing two π‘₯ plus nine twice, instead of writing it as squared. So now we need to FOIL, distribute. Two π‘₯ times two π‘₯ is four π‘₯ squared. Two π‘₯ times nine is 18π‘₯. Nine times two π‘₯ is 18π‘₯. And then nine times nine is 81. So we could keep multiplying by the 10π‘₯ plus 10.

However, let’s condense inside the pink parentheses, because we can actually combine 18π‘₯ and 18π‘₯. So now we’ve combined them, we need to FOIL again, distribute. Four π‘₯ squared times 10π‘₯ is 40π‘₯ cubed. Four π‘₯ squared times 10 is equal to 40π‘₯ squared. 36π‘₯ times 10π‘₯ is equal to 360π‘₯ squared. 36π‘₯ times 10 is equal to 360π‘₯. 81 times 10π‘₯ is equal to 810π‘₯. And 81 times 10 is equal to 810.

Now we need to combine like terms, and they will be simplified. So our highest exponent is three. We have 40π‘₯ cubed, and that should be written first. Our next highest power is π‘₯ squared. And we have 40π‘₯ squared and 360π‘₯ squared that we can combine, giving us 400π‘₯ squared. Next, our next highest power is just π‘₯ to the first power. And when we add them together, we get 1170π‘₯. And then, lastly, we have the constant of 810. So this will be our final answer.

However, we need to add feet cubed. Now instead of just writing feet cubed at the end, it kind of just looks like the feet cubed goes with 810. So if we have more than one term, it’s good to put parentheses around it so to emphasize that the entire thing is cubic feet. So this will be our final answer.

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