express the function by a formula and draw the graph, indicating that portion of the graph which has a meaning in the problem.

13. The amount of $1 at 4% simple interest, as a function of time. 
14. The volume of a sphere as a function of the radius.

15. The radius of a sphere as a function of the volume.

16. The volume of a cube as a function of the length of an edge.

Let's tackle each of the problems one by one. We'll express each function with a formula and then describe the relevant portion of the graph that has a meaningful interpretation in the problem.

### 13. The amount of \$1 at 4% simple interest, as a function of time.
**Formula:**
Simple interest can be calculated using the formula:
\[
A(t) = P(1 + rt)
\]
Where:
- $ P = 1 $ (the initial amount),
- $ r = 0.04 $ (the interest rate, 4% per year),
- $ t $ is the time in years.

Thus, the function is:
\[
A(t) = 1(1 + 0.04t) = 1 + 0.04t
\]

**Graph:**
The graph of $ A(t) = 1 + 0.04t $ is a straight line with a y-intercept of 1 and a slope of 0.04. 

**Meaningful Portion:**
The graph is meaningful for $ t \geq 0 $ because time can't be negative. The line starts at $ A(0) = 1 $ and increases as time increases.

### 14. The volume of a sphere as a function of the radius.
**Formula:**
The volume $ V $ of a sphere as a function of its radius $ r $ is given by the formula:
\[
V(r) = \frac{4}{3}\pi r^3
\]

**Graph:**
The graph of $ V(r) = \frac{4}{3}\pi r^3 $ is a cubic curve starting from the origin, increasing steeply as the radius increases.

**Meaningful Portion:**
The graph is meaningful for $ r \geq 0 $ because a radius cannot be negative. The curve starts at the origin and increases rapidly as $ r $ increases.

### 15. The radius of a sphere as a function of the volume.
**Formula:**
Given the volume $ V $, the radius $ r $ can be expressed as:
\[
r(V) = \left(\frac{3V}{4\pi}\right)^{\frac{1}{3}}
\]

**Graph:**
The graph of $ r(V) = \left(\frac{3V}{4\pi}\right)^{\frac{1}{3}} $ is a cube root function starting from the origin, slowly increasing as the volume increases.

**Meaningful Portion:**
The graph is meaningful for $ V \geq 0 $ because the volume cannot be negative. The curve starts at the origin and increases as $ V $ increases.

### 16. The volume of a cube as a function of the length of an edge.
**Formula:**
The volume $ V $ of a cube as a function of the length of an edge $ s $ is given by the formula:
\[
V(s) = s^3
\]

**Graph:**
The graph of $ V(s) = s^3 $ is a cubic curve starting from the origin, increasing steeply as the edge length increases.

**Meaningful Portion:**
The graph is meaningful for $ s \geq 0 $ because the length of an edge cannot be negative. The curve starts at the origin and increases rapidly as $ s $ increases.

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Do you need further details or have any questions?

Here are 8 related questions:

1. How does compound interest differ graphically from simple interest?
2. What is the derivative of the volume of a sphere with respect to its radius?
3. How would the graph change if the interest rate in problem 13 was higher?
4. Can you find the inverse function of the volume of a sphere concerning its radius?
5. How would the function for volume in problem 16 change if it was a rectangular prism instead of a cube?
6. What does the graph of the surface area of a sphere as a function of the radius look like?
7. How does the volume of a sphere change as its radius doubles?
8. What real-world situations would use the cube's volume formula in problem 16?

**Tip:** Understanding the physical meaning behind the variables in your functions can help guide you in selecting the meaningful portion of a graph.

Explore mathematical functions including simple interest over time, volume of a sphere as a function of its radius, radius of a sphere from its volume, and volume of a cube based on edge length. Understand the formulas and graphs associated with each function.

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