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Express
S(t) in vertex form as Valerie accelerates from 35 miles per hour to enter the highway traffic. Use 3 decimal places. Hint: convert velocity from miles per hour to miles per second.

We are given the formula for the position \( S(t) \) as a function of time:

\[
S(t) = \frac{1}{2} a t^2 + v_0 t
\]

where:
- \( a = 0.002 \, \text{miles/second}^2 \) (acceleration),
- \( v_0 \) is the initial velocity (Valerie's starting velocity).

Valerie's initial velocity is \( 35 \, \text{miles per hour} \). First, we need to convert this to miles per second. 

### Step 1: Convert \( 35 \, \text{miles/hour} \) to \( \text{miles/second} \)
There are 3600 seconds in one hour, so we convert as follows:

\[
v_0 = 35 \, \text{miles/hour} \times \frac{1 \, \text{hour}}{3600 \, \text{seconds}} = \frac{35}{3600} \, \text{miles/second} \approx 0.009722 \, \text{miles/second}
\]

### Step 2: Substitute \( a \) and \( v_0 \) into the equation for \( S(t) \)
The given acceleration \( a = 0.002 \, \text{miles/second}^2 \), and the initial velocity \( v_0 = 0.009722 \, \text{miles/second} \). Substituting these values into the formula:

\[
S(t) = \frac{1}{2} \times 0.002 \times t^2 + 0.009722 \times t
\]

Simplifying:

\[
S(t) = 0.001 t^2 + 0.009722 t
\]

### Step 3: Express \( S(t) \) in vertex form
The vertex form of a quadratic equation is generally written as:

\[
S(t) = a(t - h)^2 + k
\]

where \( (h, k) \) is the vertex of the parabola.

To convert \( S(t) = 0.001 t^2 + 0.009722 t \) to vertex form, we complete the square.

1. Factor out \( 0.001 \) from the quadratic and linear terms:

\[
S(t) = 0.001 \left( t^2 + \frac{0.009722}{0.001} t \right) = 0.001 \left( t^2 + 9.722 t \right)
\]

2. Complete the square:
   - Take half of \( 9.722 \), square it, and add it inside the parentheses.
   - Half of \( 9.722 \) is \( \frac{9.722}{2} = 4.861 \), and squaring it gives \( 4.861^2 = 23.625 \).

So we add and subtract \( 23.625 \) inside the parentheses:

\[
S(t) = 0.001 \left( t^2 + 9.722 t + 23.625 - 23.625 \right)
\]

\[
S(t) = 0.001 \left( (t + 4.861)^2 - 23.625 \right)
\]

3. Distribute the \( 0.001 \):

\[
S(t) = 0.001 (t + 4.861)^2 - 0.023625
\]

### Final Answer:
The position function in vertex form is:

\[
S(t) = 0.001 (t + 4.861)^2 - 0.023625
\]

Would you like more details on any part of this calculation, or do you have any other questions?

### 5 Related Questions:
1. How do you interpret the vertex of the quadratic function in terms of time and position?
2. What does the coefficient \( 0.001 \) in the vertex form represent in real-world terms?
3. How would the equation change if Valerie started from rest instead of 35 mph?
4. How can we calculate the time it takes for Valerie to reach a certain position using this equation?
5. What role does completing the square play in converting to vertex form?

### Tip:
Always double-check units when solving physics problems, especially when converting between different units like miles per hour and miles per second.

Express S(t) in vertex form as Valerie accelerates from 35 miles per hour to enter the highway traffic. Use 3 decimal places. Hint: convert velocity from miles per hour to miles per second.

Learn how to express the position function S(t) in vertex form as Valerie accelerates from 35 miles per hour to enter highway traffic. The solution includes converting velocity from miles per hour to miles per second and completing the square. Suitable for Grades 10-12.

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