Video: Finding an Algebraic Expression for the Volume of a Triangular Prism

Consider the shown rectangle. Write an expanded expression for its area. The given rectangle is the base of a triangular prism of height 3๐‘Ž + 1. Write an expanded expression for the volume of the prism. Simplify the expression, if possible.

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Video Transcript

Consider the shown rectangle. There are a couple of parts to this question. The first part is write an expanded expression for its area. And then, we will come on to the next part after this.

Well, in this question, what weโ€™re trying to do is find an expression for the area of our rectangle. And we know that the area of a rectangle is equal to the length multiplied by the width. So therefore, what weโ€™re gonna do is multiply the two dimensions weโ€™ve been given, which are a length of two ๐‘ฅ plus five and a width of ๐‘Ž minus two. So, if weโ€™re gonna multiply these two values together, what weโ€™re gonna do is distribute across our parentheses, or expand our brackets.

So, what this means is weโ€™re multiplying one set of parentheses by the other. And to do that, what we do is we multiply each term in the left-hand parentheses by each term in the right-hand. So, weโ€™re gonna start with two ๐‘ฅ multiplied by ๐‘Ž, which is gonna give us two ๐‘Ž๐‘ฅ. Then, weโ€™ve got two ๐‘ฅ multiplied by negative two, which is gonna give negative four ๐‘ฅ. So, thatโ€™s the two ๐‘ฅ multiplied by both terms.

Now, we move on to the positive five. So, now, weโ€™ve got positive five ๐‘Ž. And thatโ€™s because positive five multiplied by ๐‘Ž is positive five ๐‘Ž. And then, finally, what weโ€™ve got is minus 10. And thatโ€™s because positive five multiplied by negative two is negative 10. Okay, great, so, now, what we do usually is simplify this. But we canโ€™t simply this expression because we donโ€™t have any like terms.

And thatโ€™s because if we take a look, weโ€™ve got an ๐‘Ž๐‘ฅ term, an ๐‘ฅ term, an ๐‘Ž term, and then weโ€™ve got a units term. Weโ€™ve got our negative 10. So therefore, none of them are like terms because none of them have the same power of ๐‘Ž or ๐‘ฅ. The only one that does, has both ๐‘Ž and ๐‘ฅ, so we canโ€™t actually add this to any of the other terms. Right, so, now, weโ€™ve solved part A. And that is that the expanded expression for the area of our rectangle is two ๐‘Ž๐‘ฅ minus four ๐‘ฅ plus five ๐‘Ž minus 10. Okay, great, now, letโ€™s move on to the next part of the question.

So, now, for the second part of the problem, weโ€™re told that the given rectangle is the base of a triangular prism of height three ๐‘Ž plus one. Write an expanded expression for the volume of the prism. Simplify the expression, if possible.

So, what Iโ€™ve done is drawn a quick sketch of our triangular prism. So, weโ€™ve got, weโ€™ve got the base, and weโ€™ve got the measurements that we know from the first part of question. And weโ€™ve got the height, which is three ๐‘Ž plus one. Well, in the question, we were asked to find the volume. And we know that the volume of a prism is equal to the area of the cross section multiplied by the length.

And in our prism, the cross section is a triangle. So, the first thing we want to do is work out the area of the triangle. And we know that the area of a triangle is equal to half the base times the height. In this case, half the base times the perpendicular height. So, the area of the triangle is gonna be equal to a half multiplied by ๐‘Ž minus two multiplied by three ๐‘Ž plus one.

So, the first thing I do is multiply each term in the first parentheses by a half. So, we get ๐‘Ž over two minus one multiplied by three ๐‘Ž plus one. Now, if we distribute across our parentheses, we get ๐‘Ž over two multiplied by three ๐‘Ž, which gives us three ๐‘Ž squared over two. And then, we have ๐‘Ž over two multiplied by positive one, which just gives us ๐‘Ž over two. And then, negative one multiplied by three ๐‘Ž gives us negative three ๐‘Ž. And then, finally, negative one multiplied by positive one gives us negative one.

So, great, weโ€™ve got three ๐‘Ž squared over two plus ๐‘Ž over two minus three ๐‘Ž minus one. And then, what we can do is we can collect the like terms. So, weโ€™ve got positive ๐‘Ž over two, so thatโ€™s a half ๐‘Ž, minus three ๐‘Ž, which is gonna be minus two and a half ๐‘Ž or minus five over two ๐‘Ž. So, now, weโ€™ve got that our area of our cross section because itโ€™s three ๐‘Ž squared over two minus five over two ๐‘Ž minus one. So, now, what we can do is move on and find the volume of our prism.

So, to do that, what weโ€™re gonna do is multiply the area of the cross section by the length. So, weโ€™ve got three ๐‘Ž squared over two minus five over two ๐‘Ž minus one multiplied by two ๐‘ฅ plus five. Even though weโ€™ve got three terms in our left set of parentheses, it doesnโ€™t change how we distribute. We still multiply each term in the left by each term in the right.

So, weโ€™re gonna start with three ๐‘Ž squared over two multiplied by two ๐‘ฅ, which is gonna give us three ๐‘Ž squared ๐‘ฅ. And thatโ€™s because the twos will cancel cause weโ€™ve got two divided by two, which is just one. Then, weโ€™ve got three ๐‘Ž squared multiplied by ๐‘ฅ, which gives us three ๐‘Ž squared ๐‘ฅ. And then, this is gonna be plus 15 over two ๐‘Ž squared. And then, weโ€™ve got minus five ๐‘Ž๐‘ฅ. And thatโ€™s cause you got negative five over two ๐‘Ž multiplied by two ๐‘ฅ. Well, again, the twos would cancel out, so weโ€™re left with negative five ๐‘Ž๐‘ฅ. Then, weโ€™ve got minus 25 over two ๐‘Ž.

And then, we move on to the negative one. Weโ€™ve got negative one multiplied by two ๐‘ฅ is negative two ๐‘ฅ. And then, finally, weโ€™ve got negative one multiplied by positive five, which gives us negative five. Well, now, if we check back to the question, it says simplify the expression, if possible. Well, can we simplify this? Well, the answer is no. And thatโ€™s again because we donโ€™t have any like terms.

Because weโ€™ve got an ๐‘Ž squared ๐‘ฅ term, an ๐‘Ž squared term, an ๐‘Ž๐‘ฅ term, an ๐‘Ž term, an ๐‘ฅ term, and just a unit term. So, none of them are like terms. So therefore, we can say that the expression for the volume of the prism is three ๐‘Ž squared ๐‘ฅ plus 15 over two ๐‘Ž squared minus five ๐‘Ž๐‘ฅ minus 25 over two ๐‘Ž minus two ๐‘ฅ minus five.

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