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Given that π₯ is proportional to π¦ cubed and π₯ equals 81 when π¦ is equal to three, what is π₯ when π¦ is equal to four?
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In this question, weβre dealing with direct proportion or variation.
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We are told that π₯ varies directly with π¦ cubed.
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This can be rewritten as the equation π₯ is equal to the constant π multiplied by π¦ cubed.
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Dividing both sides of this equation by π¦ cubed gives us the constant π is equal to π₯ divided by π¦ cubed.
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We are also told that when π₯ is equal to 81, π¦ is equal to three.
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This means that we can calculate the value of π by dividing 81 by three cubed.
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Three cubed is equal to 27, as three multiplied by three is nine and multiplying this by three gives us 27.
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This means that π is equal to 81 divided by 27.
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There are three 27s in 81.
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Therefore, π is equal to three.
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If we didnβt spot this, we could have firstly canceled the fraction by dividing the numerator and denominator by nine.
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This would leave us with nine divided by three, which we know is equal to three.
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Alternatively, we might have noticed that 81 is equal to three to the fourth power.
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And dividing this by three to the third power or three cubed would give us three to the power of one, which is three.
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Substituting this value of π back into our equation gives us π₯ is equal to three π¦ cubed.
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We now need to calculate the value of π₯ when π¦ is equal to four.
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This gives us π₯ is equal to three multiplied by four cubed.
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Four cubed is equal to 64.
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And multiplying this by three gives us 192.
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If π₯ is proportional to π¦ cubed and π₯ equals 81 when π¦ is equal to three, then π₯ is equal to 192 when π¦ is equal to four.