Question

Use the table to answer the question.

Cake Area Versus Cake Pan Radius
Cake Area (in.2
) Cake Pan Radius
(in.)
0 0
25 2.8
40 3.54
55 4.15
70 4.69
The table shows the radius of a cake pan as a function of the cake’s base area. Use technology to create a square root function that models this data. Round values to the nearest hundredth.

(1 point)
The square root function that fits the table of data is r(x)=
x−−√

Answer 1

To create a square root function that models the relationship between cake area and cake pan radius based on the data provided, you can use the general form of a square root function:

\[ r(x) = a \sqrt{x} + b \]

Where:
- \( r(x) \) is the radius of the pan,
- \( x \) is the area of the cake, and
- \( a \) and \( b \) are constants to be determined.

From the data, we can use points to find the values of \( a \) and \( b \). However, considering the behavior of the data, we can simplify our approach by estimating a function that captures the relationship.

Looking at the provided data points:

- When \( x = 0 \), \( r(x) = 0 \).
- For other values, we observe a pattern that suggests a square root relationship.

Let’s apply a square root function and analyze it. A good start is to use the first two data points to find constants.

1. Start with the points \( (25, 2.8) \) and \( (40, 3.54) \):
- For the point \( (25, 2.8) \):
\[
2.8 = a \sqrt{25} + b \Rightarrow 2.8 = 5a + b
\]
- For the point \( (40, 3.54) \):
\[
3.54 = a \sqrt{40} + b \Rightarrow 3.54 = a \cdot 6.32 + b
\]

You can solve these two equations simultaneously to find \( a \) and \( b \).

2. Set up the equations:
1. \( b = 2.8 - 5a \) — (1)
2. Substitute (1) into the second equation:
\[
3.54 = a \cdot 6.32 + 2.8 - 5a
\]
\[
3.54 - 2.8 = (6.32 - 5)a
\]
\[
0.74 = 1.32a \Rightarrow a = \frac{0.74}{1.32} \approx 0.56
\]

3. Substitute back to find \( b \):
- From equation (1):
\[
b = 2.8 - 5(0.56) \approx 2.8 - 2.8 = 0
\]

Thus, the function can be approximated as:
\[
r(x) = 0.56 \sqrt{x}
\]

To round the coefficient to the nearest hundredth, we get:
\[
r(x) \approx 0.56 \sqrt{x}
\]

So the square root function that fits the table of data is:

\[
\boxed{r(x) = 0.56 \sqrt{x}}
\]