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Instructions:

1. Read the question carefully.
2. Make sure to answer all parts of the question before you complete the assignment.
3. Here are a few resources to help you with similar questions:
i. [http://num.escallonweb.com/Halls_acucalciminimumspeed.htm](http://num.escallonweb.com/Halls_acucalciminimumspeed.htm)
ii. [http://mistermartin.net/texts/s_math/financial_algebra_textbook/5.9.pdf](http://mistermartin.net/texts/s_math/financial_algebra_textbook/5.9.pdf)

In the assignment, you will be using Desmos. If you are unfamiliar with Desmos, here is a resource page for you.

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Skid Marks:

When there is a vehicle accident, the police are called in to investigate. One part of the investigation is to estimate the speed the car was traveling at the time of the accident. Using the length of skid marks on the road, police can use a formula to determine, within reasonable error, the minimum speed of the car at the time.

The formula is:
[tex]\[ S = \sqrt{30df} \][/tex]
(Note that the formula ignores brake efficiency.)

Where:
- [tex]\( S \)[/tex] represents the speed of the car in mph
- [tex]\( d \)[/tex] represents the distance the car skidded in feet
- [tex]\( f \)[/tex] represents the coefficient of friction between the road and tire. For simplicity, [tex]\( f \)[/tex] is typically either about 1.0 if the road was dry or 0.5 if the road was wet.

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1. Rewrite the formula using a fractional exponent instead of a radical.

2. Fill in the table below for the estimated speed of the car based on the length of skid marks. It was a clear and dry day. Show all your calculations for the table answers. Round answers to the nearest hundredth.

[tex]\[
\begin{array}{|l|l|}
\hline
\text{Skid Mark Lengths (feet)} & \text{Speed of Car (mph)} \\
\hline
10 \text{ ft} & \\
\hline
25 \text{ ft} & \\
\hline
40 \text{ ft} & \\
\hline
70 \text{ ft} & \\
\hline
130 \text{ ft} & \\
\hline
\end{array}
\][/tex]



Answer :

Sure, let's solve this problem step-by-step.

### 1. Rewrite the Formula as a Fraction Exponent

The given formula is:
[tex]\[ S = \sqrt{30 \cdot d \cdot f} \][/tex]

We know that a square root can be represented as a fractional exponent. Specifically, the square root of a variable [tex]\(x\)[/tex] can be written as [tex]\(x^{1/2}\)[/tex]. Therefore, rewriting the formula, we get:
[tex]\[ S = (30 \cdot d \cdot f)^{1/2} \][/tex]

### 2. Estimate the Speed of the Car Based on the Length of Skid Marks

We need to fill in the table for the estimated speed of the car based on the length of skid marks. We are given that it was a clear and dry day, so the coefficient of friction ([tex]\(f\)[/tex]) is 1.0.

#### Skid Mark Lengths and Corresponding Speeds:
1. 10 feet:
[tex]\[ S = (30 \cdot 10 \cdot 1.0)^{1/2} \][/tex]
[tex]\[ S = (300)^{1/2} \][/tex]
[tex]\[ S = \sqrt{300} \approx 17.32 \, \text{mph} \][/tex]

2. 25 feet:
[tex]\[ S = (30 \cdot 25 \cdot 1.0)^{1/2} \][/tex]
[tex]\[ S = (750)^{1/2} \][/tex]
[tex]\[ S = \sqrt{750} \approx 27.39 \, \text{mph} \][/tex]

3. 40 feet:
[tex]\[ S = (30 \cdot 40 \cdot 1.0)^{1/2} \][/tex]
[tex]\[ S = (1200)^{1/2} \][/tex]
[tex]\[ S = \sqrt{1200} \approx 34.64 \, \text{mph} \][/tex]

4. 70 feet:
[tex]\[ S = (30 \cdot 70 \cdot 1.0)^{1/2} \][/tex]
[tex]\[ S = (2100)^{1/2} \][/tex]
[tex]\[ S = \sqrt{2100} \approx 45.83 \, \text{mph} \][/tex]

5. 130 feet:
[tex]\[ S = (30 \cdot 130 \cdot 1.0)^{1/2} \][/tex]
[tex]\[ S = (3900)^{1/2} \][/tex]
[tex]\[ S = \sqrt{3900} \approx 62.45 \, \text{mph} \][/tex]

Now, we can fill in the table with these calculated speeds:

\begin{tabular}{|l|l|}
\hline Skid Mark Lengths (feet) & Speed of Car (mph) \\
\hline 10 ft & 17.32 \\
\hline 25 ft & 27.39 \\
\hline 40 ft & 34.64 \\
\hline 70 ft & 45.83 \\
\hline 130 ft & 62.45 \\
\hline
\end{tabular}