Question

use the table to answer the question.
capacity of rain barrel and corresponding radius
capacity of rain barrel (l) radius of rain barrel (cm)
150 23.16
200 26.75
250 29.90
300 32.76
350 35.38
a factory produces many different rain barrels. all the rain barrels are the same height, but their radii vary. the table shows the radius of a rain barrel as a function of its capacity. use technology to create a square root function that fits the table of data.
(1 point)
the radius of a rain barrel as a function of its capacity is modeled by the function $r(x)=0.53\sqrt{x}$.
the radius of a rain barrel as a function of its capacity is modeled by the function $r(x)=\sqrt{x}+10.91$.
the radius of a rain barrel as a function of its capacity is modeled by the function $r(x)=1.89\sqrt{x}$.
the radius of a rain barrel as a function of its capacity is modeled by the function $r(x)=\sqrt{x}+14.09$.

Explanation:

Step1: Test each function with data points

Take the first data - point where capacity $x = 150$ and radius $r=23.16$.

Step2: Test function $r(x)=0.53\sqrt{x}$

Substitute $x = 150$ into $r(x)=0.53\sqrt{x}$, we get $r(150)=0.53\sqrt{150}\approx0.53\times12.25\approx6.49$. This is not close to $23.16$.

Step3: Test function $r(x)=\sqrt{x}+10.91$

Substitute $x = 150$ into $r(x)=\sqrt{x}+10.91$, we get $r(150)=\sqrt{150}+10.91\approx12.25 + 10.91=23.16$.

Step4: Test function $r(x)=1.89\sqrt{x}$

Substitute $x = 150$ into $r(x)=1.89\sqrt{x}$, we get $r(150)=1.89\sqrt{150}\approx1.89\times12.25\approx23.15$.

Step5: Test function $r(x)=\sqrt{x}+14.09$

Substitute $x = 150$ into $r(x)=\sqrt{x}+14.09$, we get $r(150)=\sqrt{150}+14.09\approx12.25+14.09 = 26.34$. This is not close to $23.16$.
We can further test with other data - points. For example, when $x = 200$:
For $r(x)=\sqrt{x}+10.91$, $r(200)=\sqrt{200}+10.91\approx14.14+10.91 = 25.05$.
For $r(x)=1.89\sqrt{x}$, $r(200)=1.89\sqrt{200}\approx1.89\times14.14\approx26.73$ which is closer to the value of $26.75$ in the table.

Answer:

The radius of a rain barrel as a function of its capacity is modeled by the function $r(x)=1.89\sqrt{x}$.