Video: AQA GCSE Mathematics Higher Tier Pack 2 β€’ Paper 1 β€’ Question 24

A cylinder has a length of π‘₯ m and the radius of its base is π‘Ÿ m. The length π‘₯ is increased by 20%. The radius π‘Ÿ is decreased by 40%. Calculate and describe the percentage change in the volume of the cylinder.

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Video Transcript

A cylinder has a length of π‘₯ meters and the radius of its base is π‘Ÿ meters. The length π‘₯ is increased by 20 percent. The radius π‘Ÿ is decreased by 40 percent. Calculate and describe the percentage change in the volume of the cylinder.

Remember the formula for volume of a prism is the area of that prism’s cross section multiplied by its length. The cross section of the cylinder is a circle. And the formula for area of a circle with a radius of π‘Ÿ is πœ‹π‘Ÿ squared.

So the volume of a cylinder is πœ‹ multiplied by its radius squared multiplied by its length. We can use this formula to calculate the volume of the cylinder before the length and radius have changed.

The area of the cross section is πœ‹π‘Ÿ squared and its length is π‘₯. So the volume of the original cylinder is πœ‹π‘Ÿ squared multiplied by π‘₯. Its length π‘₯ is then increased by 20 percent.

Since the original length will be 100 percent and we’re increasing it by 20 percent, that means we now have 120 percent of the original length. As a decimal, 120 percent is equivalent to 1.2. Because to change from a percentage into a decimal, do we divide by 100. And 120 divided by 100 is 1.2.

This means the new length can be found by multiplying the original by 1.2; that’s 1.2π‘₯.

We perform a similar process to find the new radius of the cylinder. This time, it’s been decreased by 40 percent. 100 minus 40 is 60. So the new radius is 60 percent the length of the old radius. 60 percent is equivalent to 0.6 since 60 divided by 100 is 0.6.

This means we can find the radius of the new cylinder by multiplying the radius of the original cylinder by 0.6. That’s 0.6π‘Ÿ. And we can substitute these values into the formula for the volume.

This time, πœ‹ multiplied by the radius squared is πœ‹ multiplied by 0.6π‘Ÿ squared. And the length is 1.2π‘₯. 0.6π‘Ÿ all squared is the same as 0.6 squared multiplied by π‘Ÿ squared.

And since multiplication is commutative, that means we can do it in any order. We can rewrite this slightly as 0.6 squared multiplied by 1.2 squared multiplied by πœ‹π‘Ÿ squared π‘₯. 0.6 squared multiplied by 1.2 is the same as 0.432.

And we now have expressions for the volume of the original cylinder and the new cylinder. We’re not done though. We need to calculate and describe the percentage change in the volume of the cylinder.

The formula we can use for percentage change is to find the change itself, divide that by the original, and then multiply it by 100. The change is the new minus the old; that’s 0.432 πœ‹π‘Ÿ squared π‘₯ minus πœ‹π‘Ÿ squared π‘₯.

And we can factorise this expression. This will make our life easier in a moment by taking out a common factor of πœ‹π‘Ÿ squared π‘₯. Doing so and we get πœ‹π‘Ÿ squared π‘₯ multiplied by 0.432 minus one. 0.432 minus one is negative 0.568. So the change is negative 0.568 πœ‹π‘Ÿ squared π‘₯.

Let’s substitute this into the formula for percentage change. It’s negative 0.568 πœ‹π‘Ÿ squared π‘₯ over the original, which we worked out to be πœ‹π‘Ÿ squared π‘₯. And then, we multiply that by 100.

Notice how both the numerator and the denominator of this fraction have a common factor of πœ‹π‘Ÿ squared π‘₯. So we’re left with negative 0.568 multiplied by 100, which is negative 56.8 percent.

A negative percentage change means it’s decreased in volume. This is a 56.8 percent decrease.

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