 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.
 18.  The surface area of a cube as a function of the volume of a cube.
 17. The surface area of a cube as a function of the length of an edge.

Let's address each of these functions one by one, express them as formulas, and describe the corresponding graphs, including the portions that have meaning within the context of each problem.

### 13. The Amount of $1 at 4% Simple Interest as a Function of Time

**Formula:**

The simple interest formula is given by:

\[
A(t) = P(1 + rt)
\]

Where:
- $ A(t) $ is the amount after time $ t $.
- $ P = 1 $ is the principal amount (initial amount).
- $ r = 0.04 $ is the interest rate (4%).

So the formula becomes:

\[
A(t) = 1(1 + 0.04t) = 1 + 0.04t
\]

**Graph:**
- This is a linear function, so the graph will be a straight line.
- The x-axis represents time $ t $, and the y-axis represents the amount $ A(t) $.
- The portion of the graph that has meaning in this context is for $ t \geq 0 $ since time cannot be negative.

### 14. The Volume of a Sphere as a Function of the Radius

**Formula:**

The volume $ V $ of a sphere as a function of the radius $ r $ is given by:

\[
V(r) = \frac{4}{3}\pi r^3
\]

**Graph:**
- This is a cubic function, so the graph will be a curve starting from the origin (0,0) and increasing rapidly.
- The x-axis represents the radius $ r $, and the y-axis represents the volume $ V(r) $.
- The portion of the graph that has meaning is for $ r \geq 0 $ since the radius cannot be negative.

### 17. The Surface Area of a Cube as a Function of the Length of an Edge

**Formula:**

The surface area $ S $ of a cube as a function of the length of an edge $ s $ is given by:

\[
S(s) = 6s^2
\]

**Graph:**
- This is a quadratic function, so the graph will be a parabola.
- The x-axis represents the edge length $ s $, and the y-axis represents the surface area $ S(s) $.
- The meaningful portion of the graph is for $ s \geq 0 $ since the edge length cannot be negative.

### 18. The Surface Area of a Cube as a Function of the Volume of a Cube

**Formula:**

First, express the volume $ V $ of the cube as a function of the edge length $ s $:

\[
V(s) = s^3 \quad \Rightarrow \quad s = V^{\frac{1}{3}}
\]

Now, substitute $ s = V^{\frac{1}{3}} $ into the surface area formula:

\[
S(V) = 6(V^{\frac{1}{3}})^2 = 6V^{\frac{2}{3}}
\]

**Graph:**
- This is a power function with an exponent of $ \frac{2}{3} $, so the graph will be a curve that starts from the origin and increases, but not as rapidly as a cubic function.
- The x-axis represents the volume $ V $, and the y-axis represents the surface area $ S(V) $.
- The portion of the graph that has meaning is for $ V \geq 0 $ since the volume cannot be negative.

---

Would you like more details on any of these functions or their graphs? Here are some related questions you might find interesting:

1. How would the graph of the amount at simple interest change with different interest rates?
2. What is the effect of compounding interest on the graph of the amount as a function of time?
3. How does the surface area of a sphere change as a function of its radius?
4. What is the relationship between the surface area and volume of a sphere?
5. How does the volume of a cube change as a function of its surface area?
6. How does the graph of a quadratic function differ from a cubic function?
7. What happens to the surface area of a cube if the edge length is doubled?
8. How do these functions differ in terms of their rate of change?

**Tip:** When graphing functions, always consider the physical or practical constraints of the variables, such as time being non-negative or a length that cannot be negative. This helps focus on the relevant portions of the graph.

Explore mathematical functions including simple interest over time, sphere volume based on radius, and cube surface area in relation to edge length and volume. Learn formulas and graphs for these functions. Suitable for high school students.

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