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Chapter 7: Problem 2

In parts (a)-(c), evaluate the functions at \(R\) and \(3 R\). a. \(C(r)=2 \pi r\), the circumference of a circle with radius \(r\) b. \(A(r)=\pi r^{2},\) the area of a circle with radius \(r\) c. \(V(r)=\frac{4}{3} \pi r^{3},\) the volume of a sphere with radius \(r\) d. Describe what happens to \(C(r), A(r)\), and \(V(r)\) when the radius triples from \(R\) to \(3 R\).

Short Answer

Expert verified
Circumference multiplies by 3, area by 9, and volume by 27 when radius triples.

Step by step solution

01

Evaluate C(r) at R

The function for the circumference of a circle is given by: \( C(r) = 2 \pi r \). Substitute \( r = R \): \( C(R) = 2 \pi R \).
02

Evaluate C(r) at 3R

Substitute \( r = 3R \) into the function. \( C(3R) = 2 \pi (3R) = 6 \pi R \).
03

Evaluate A(r) at R

The function for the area of a circle is given by: \( A(r) = \pi r^2 \). Substitute \( r = R \): \( A(R) = \pi R^2 \).
04

Evaluate A(r) at 3R

Substitute \( r = 3R \) into the function. \( A(3R) = \pi (3R)^2 = 9 \pi R^2 \).
05

Evaluate V(r) at R

The function for the volume of a sphere is given by: \( V(r) = \frac{4}{3} \pi r^3 \). Substitute \( r = R \): \( V(R) = \frac{4}{3} \pi R^3 \).
06

Evaluate V(r) at 3R

Substitute \( r = 3R \) into the function. \( V(3R) = \frac{4}{3} \pi (3R)^3 = \frac{4}{3} \pi (27R^3) = 36 \pi R^3 \).
07

Describe behavior of functions when radius triples

When the radius is tripled from \( R \) to \( 3R \): \( C(r) \) becomes 3 times larger (from \( 2 \pi R \) to \( 6 \pi R \)). \( A(r) \) becomes 9 times larger (from \( \pi R^2 \) to \( 9 \pi R^2 \)). \( V(r) \) becomes 27 times larger (from \( \frac{4}{3} \pi R^3 \) to \( 36 \pi R^3 \)).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Circumference of a Circle
The circumference of a circle is the distance around the circle. You can calculate it using the formula: \( C(r) = 2 \pi r \) . Here, \pi (pi) is a constant approximately equal to 3.14, and \( r \) is the radius of the circle.
Let's say you have a circle with a radius \( R \). Substituting \( R \) into the formula gives us: \( C(R) = 2 \pi R \).
Now, if the radius triples to \( 3R \), the new circumference will be: \( C(3R) = 2 \pi (3R) = 6 \pi R \).
This shows that when the radius is tripled, the circumference also triples. Understanding these relationships helps in various practical applications such as designing wheels or calculating travel distances.
Area of a Circle
The area of a circle is the amount of space enclosed within its boundary. The formula to find the area is: \( A(r) = \pi r^2 \).
Again, \( \pi \approx 3.14 \) and \( r \) is the radius. For a circle with radius \( R \), substituting \( R \) into the formula gives us: \( A(R) = \pi R^2 \).
If the radius triples to \( 3R \), the new area will be: \( A(3R) = \pi (3R)^2 = 9 \pi R^2 \).
This shows that when the radius is tripled, the area becomes nine times larger. This concept is very important in real-life scenarios like designing gardens or determining the surface area of circular objects.
Volume of a Sphere
The volume of a sphere measures the amount of space it occupies. You can calculate it using the formula: \( V(r) = \frac{4}{3} \pi r^3 \).
Here, \pi is approximately 3.14, and \( r \) is the radius of the sphere. For a sphere with radius \( R \), substituting \( R \) into the formula gives us: \( V(R) = \frac{4}{3} \pi R^3 \).
If the radius triples to \( 3R \), the new volume will be: \( V(3R) = \frac{4}{3} \pi (3R)^3 = 36 \pi R^3 \).
When the radius is tripled, the volume becomes twenty-seven times larger. This scaling is important in fields such as packing, manufacturing, and even in science when dealing with spherical cells or bubbles.
Scaling of Geometric Quantities
Scaling refers to how geometric quantities change when the size of a shape is increased or decreased. When you scale a shape by a factor, you multiply its dimensions by that factor.
For example, let鈥檚 consider the radius of a circle or sphere:
  • If the radius is doubled, the circumference (which is linear) doubles.
  • The area (which depends on the square of the radius) becomes four times larger.
  • The volume (which depends on the cube of the radius) becomes eight times larger.
In the exercise, we looked at tripling the radius:
  • Circumference becomes three times larger.
  • Area becomes nine times larger.
  • Volume becomes twenty-seven times larger.
Understanding how different dimensions scale helps in problem-solving and practical applications, making complex geometric problems more intuitive.

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Most popular questions from this chapter

Assume \(Y\) is directly proportional to \(X^{3}\). a. Express this relationship as a function where \(Y\) is the dependent variable. b. If \(Y=10\) when \(X=2\), then find the value of the constant of proportionality in part (a). c. If \(X\) is increased by a factor of 5 , what happens to the value of \(Y ?\) d. If \(X\) is divided by \(2,\) what happens to the value of \(Y ?\) e. Rewrite your equation from part (a), solving for \(X .\) Is \(X\) directly proportional to \(Y ?\)

(Graphing program optional.) Evaluate each of the following functions at \(0,0.5,\) and \(1 .\) Then, on the same grid, graph each over the interval [0,1] . Compare the graphs. a. \(y_{1}=x \quad y_{2}=x^{1 / 2} \quad y_{3}=x^{1 / 3} \quad y_{4}=x^{1 / 4}\) b. \(y_{5}=x^{2} \quad y_{6}=x^{3} \quad y_{7}=x^{4}\)

A box with a lid has a square base with each base side \(x\) inches and a height of \(h\) inches. a. Write the formula for the volume of the box. b. Write the formula for the surface area. (Hint: The surface area of the box is the sum of the areas of each side of the box.) c. If the side of the base is tripled and the height remains constant, by what factor is the volume increased? d. If the side of the base triples (from \(x\) to \(3 x\) ) with \(h\) held constant, what is the change in the surface area?

a. \(B\) is inversely proportional to \(x^{4}\). What is the effect on \(B\) of doubling \(x\) ? b. \(Z\) is inversely proportional to \(x^{p}\), where \(p\) is a positive integer. What is the effect on \(Z\) of doubling \(x\) ?

The intensity of light from a point source is inversely proportional to the square of the distance from the light source. If the intensity is 4 watts per square meter at a distance of \(6 \mathrm{~m}\) from the source, find the intensity at a distance of \(8 \mathrm{~m}\) from the source. Find the intensity at a distance of \(100 \mathrm{~m}\) from the source.
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