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Choose one of the following scenarios, explain your reasoning

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Choose one of the following scenarios, explain your reasoning and provide an example that demonstrates your logic.
Ø What happens to the circumference of a circle if you double the radius? What happens if you double the diameter? What happens if you triple the radius?
Ø What happens to the area of a circle if you double the radius? What happens if you double the diameter? What happens if you triple the radius?
Ø What is the effect on the area of a triangle if the base is doubled and the height is cut in half? What happens to the area, if the base is doubled and the height remains the same?

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Q) What happens to the circumference of a circle if you double the radius? What happens if you double the diameter? What happens if you triple the radius?

SOLUTION:

Circumference = 2πr

- Let r1 be a radius, then circumference c1 is given as

c1 = 2πr1

If r2 = 2r1 (radius doubled), then

c2 = 2πr2 = 2π(2r1) = 2(2πr1) = 2c1

Hence when radius is doubled, circumference is doubled.

- In terms of diameter d, circumference is given as:

circumference = πd

For diameter d1, circumference c1 is given as

c1 = πd1

If d2 = 2d1, then

c2 = πd2 = π(2d1) = 2πd1 = 2c1

Hence when the diameter is doubled, circumference doubles.

If r3 = 3r1, then

c3 = 2πr3 = 2π(3r1) = 3 (2πr1) = 3c1

Hence, when radius is tripled, circumference triples.

Q) What happens to the area of a circle if you double the radius? What happens if you double the diameter? What happens if you triple the radius?

SOLUTION:

Area = π (radius)^2

If r1 is radius, then

A1 = π (r1)^2

If r2 = 2r1, then

A2 = π (r2)^2 = π (2r1)^2 = 4 π (r1)^2 = 4 A1

If radius is doubled, then area is quadrupled.

In terms of diameter, area of circle is given as

Area = π (d/2)^2 = (π/4) d^2

Let d1 be a diameter, then

A1 = (π/4) (d1)^2

Let d2 = 2d1, then

A2 = (π/4)(d2)^2 = (π/4)(2d1)^2 = 4(π/4)(d1)^2 = 4A1

Hence, when the diameter is doubled, area is quadrupled.

Now, if r3 = 3r1, then

A3 = π(r3)^2 = π(3r1)^2 = 9 π(r1)^2 = 9A1

Hence, when the radius is tripled, area increases 9 times.


Q) What is the effect on the area of a triangle if the base is doubled and the height is cut in half? What happens to the area, if the base is doubled and the height remains the same?

SOLUTION:

Area of triangle = (1/2)(base)(height)

Let b1 and h1 be heights, then

A1 = (1/2)(b1)(h1)

Now if b2 = 2b1 and h2 = (h1/2)

Then

A2 = (1/2)(b2)(h2) = (1/2)(2b1)(h1/2) = (1/2)(b1)(h1) = A1

Hence when the base is doubled and height is halved, area of the triangle remains the same.

Now if b3 = 2b1 and h3 = h1, then

A3 = (1/2)(b3)(h3) = (1/2)(2b1)(h1) = 2 (1/2)(b1)(h1) = 2A1

That shows that when base is doubled but the height remains the same, area of the triangle is doubled.

namstech and other Math Homework Specialists are ready to help you
Customer: replied 5 years ago.
where do the "4" and "9" come from in your area formulas?
You are probably referring to the following part:

If r2 = 2r1, then
A2 = π (r2)^2 = π (2r1)^2 = 4 π (r1)^2 = 4 A1

Here, new radius r2 is double than that of radius r1. Now when you put that value in the formula, we get square of the value 2r1, whose square is 4r1^2 (i.e. square of 2 and square of r1^2). That is how we get 4 in the formula. Same argument goes for the radius r3 that is 3 times r1 i.e. 3r1^2. Squaring it in the formula yields the value 9r1^2 and hence we have 9 in the area.

I hope this helps. If there is still some confusion, please don't hesitate to ask.

Regards

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