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namstech, Electrical Engineer
Category: Math Homework
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Experience:  M.S. Signals and Information Processing
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Choose one of the following scenarios, explain your reasoning

Customer Question

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?
Submitted: 3 years ago.
Category: Math Homework
Expert:  namstech replied 3 years ago.

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:

A1 = π (r1)^2

If r2 = 2r1, then

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

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, Electrical Engineer
Category: Math Homework
Satisfied Customers: 2126
Experience: M.S. Signals and Information Processing
Customer: replied 3 years ago.
where do the "4" and "9" come from in your area formulas?
Expert:  namstech replied 3 years ago.
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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