Jake owns two gas stations in town. The projected revenue of the first gas station for the next 12 months is

0.3*t*2 + 150*t*    (0 ≤ *t* ≤ 12) 

thousand dollars *t* months from now. The projected revenue of the second gas station for the next 12 months is

0.8*t*2 + 300*t*    (0 ≤ *t* ≤ 12) 

thousand dollars *t* months from now. Find an expression that gives the amount by which the revenue of the second gas station will exceed that of the first gas station in month *t* (0 ≤ *t* ≤ 12).

We are given the projected revenue of two gas stations for the next 12 months, and we need to find an expression for the amount by which the revenue of the second gas station exceeds that of the first gas station in month *t*, where \(0 \leq t \leq 12\).

### Step 1: Write down the revenue functions

- For the first gas station, the revenue function is:
  \[
  R_1(t) = 0.3t^2 + 150t \quad \text{(in thousand dollars)}
  \]
  
- For the second gas station, the revenue function is:
  \[
  R_2(t) = 0.8t^2 + 300t \quad \text{(in thousand dollars)}
  \]

### Step 2: Find the difference between the revenues

The goal is to find an expression for how much the revenue of the second gas station exceeds that of the first gas station. To do this, subtract \(R_1(t)\) from \(R_2(t)\):

\[
\Delta R(t) = R_2(t) - R_1(t)
\]

Substituting the expressions for \(R_2(t)\) and \(R_1(t)\):

\[
\Delta R(t) = (0.8t^2 + 300t) - (0.3t^2 + 150t)
\]

### Step 3: Simplify the expression

Now simplify the expression:

\[
\Delta R(t) = 0.8t^2 + 300t - 0.3t^2 - 150t
\]

\[
\Delta R(t) = (0.8t^2 - 0.3t^2) + (300t - 150t)
\]

\[
\Delta R(t) = 0.5t^2 + 150t
\]

### Final Answer:
The amount by which the revenue of the second gas station exceeds that of the first gas station in month *t* is given by the expression:

\[
\Delta R(t) = 0.5t^2 + 150t \quad \text{(in thousand dollars)}
\]

Would you like further details on the interpretation or application of this expression?

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Here are 5 related questions for further practice:
1. What is the revenue of each gas station at \(t = 6\) months?
2. At which month *t* will the difference in revenues be the greatest?
3. What is the combined revenue of both gas stations at \(t = 12\) months?
4. How does the difference in revenues change with increasing *t* values?
5. Can the difference in revenues be negative for any *t* between 0 and 12?

**Tip:** When simplifying expressions, always group like terms (such as \(t^2\) and \(t\)) to make calculations easier.

Jake owns two gas stations in town. The projected revenue of the first gas station for the next 12 months is 0.3*t*2 + 150*t* (0 ≤ *t* ≤ 12) thousand dollars *t* months from now. The projected revenue of the second gas station for the next 12 months is 0.8*t*2 + 300*t* (0 ≤ *t* ≤ 12) thousand dollars *t* months from now. Find an expression that gives the amount by which the revenue of the second gas station will exceed that of the first gas station in month *t* (0 ≤ *t* ≤ 12).

Find how much the revenue of one gas station exceeds the other with a quadratic function. The revenue functions involve quadratic expressions, perfect for Grades 9-12. Explore step-by-step calculations with key concepts in algebra.

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