Suppose ​$5500 is invested in a savings account for 10 years ​(120 ​months), with an annual interest rate of​ r, compounded monthly. The amount of money in the account after 10 years is Upper A left parenthesis r right parenthesis equals 5500 left parenthesis 1 plus StartFraction r Over 12 EndFraction right parenthesis Superscript 120
.
a. Use the Intermediate Value Theorem to show there is a value of r in ​(0,0.08)long dashan interest rate between​ 0% and ​8%long dashthat allows you to reach your savings goal of​ $7000 in 10 years.
b. Use a graph to illustrate your explanation in part​ (a); then approximate the interest rate required to reach your goal.
Question content area bottom
Part 1
a. The Intermediate Value Theorem states that if f is 
continuous
 on the interval 
[a,b]
 and L is a number 
strictly between
 ​f(a) and​ f(b), then 
there exists at least one
 number c in 
(a,b)
 satisfying 
f left parenthesis c right parenthesis equals L.
Part 2
For what values of r is A​ continuous?
A.
A is continuous for all r.
Your answer is correct.B.
A is continuous only on​ (0,0.08).
C.
A is continuous only on​ [0,0.08].
D.
A is not continuous on any interval.
Part 3
Find​ A(0).
​A(0)equals
  
5500 ​(Type an integer or a​ decimal.)
Part 4
Find​ A(0.08).
​A(0.08)almost equals
  
enter your response here ​(Round to the nearest integer as​ needed.)

Let's break this down step by step.

### Part 1: Use the Intermediate Value Theorem (IVT)

The Intermediate Value Theorem (IVT) states that if a function $ f(x) $ is continuous on the interval $[a, b]$ and $ N $ is any number between $ f(a) $ and $ f(b) $, then there exists at least one number $ c $ in the interval $(a, b)$ such that $ f(c) = N $.

For the problem at hand, the function is:
\[
A(r) = 5500 \left(1 + \frac{r}{12}\right)^{120}
\]
This function $ A(r) $ gives the amount of money in the account after 10 years, depending on the annual interest rate $ r $.

We need to show that there is a value of $ r $ in $ (0, 0.08) $ that allows $ A(r) $ to reach $7000.

1. **Continuity**: The function $ A(r) $ is continuous for all $ r $ because it is a composition of continuous functions (a linear function inside an exponential function).

2. **Values of $ A(r) $ at the endpoints**:
   - At $ r = 0 $:
     \[
     A(0) = 5500 \left(1 + \frac{0}{12}\right)^{120} = 5500
     \]
   - At $ r = 0.08 $:
     \[
     A(0.08) = 5500 \left(1 + \frac{0.08}{12}\right)^{120}
     \]

     We’ll calculate this later in Part 4.

3. Since $ A(0) = 5500 $ and we are seeking $ A(r) = 7000 $ and $ A(r) $ is continuous, and $ A(0) < 7000 $, if $ A(0.08) $ is greater than $ 7000 $, then by the IVT, there must be some $ r $ in the interval $ (0, 0.08) $ where $ A(r) = 7000 $.

### Part 2: For what values of $ r $ is $ A(r) $ continuous?

**Answer**: 
- $ A(r) $ is continuous for all $ r $. The correct answer is **A**.

### Part 3: Find $ A(0) $.

We already calculated this:
\[
A(0) = 5500
\]

### Part 4: Find $ A(0.08) $.

Let's compute $ A(0.08) $:

\[
A(0.08) = 5500 \left(1 + \frac{0.08}{12}\right)^{120}
\]
\[
A(0.08) = 5500 \left(1 + 0.0066667\right)^{120} \approx 5500 \times (1.0066667)^{120}
\]

Now, calculating the above:

\[
(1.0066667)^{120} \approx 2.21964
\]

\[
A(0.08) \approx 5500 \times 2.21964 \approx 12208
\]

So, $ A(0.08) $ is approximately 12208 (rounded to the nearest integer).

### Summary

- **Part 1a**: By IVT, since $ A(r) $ is continuous and $ 5500 < 7000 < 12208 $, there is a value $ r \in (0, 0.08) $ such that $ A(r) = 7000 $.
- **Part 2**: $ A(r) $ is continuous for all $ r $.
- **Part 3**: $ A(0) = 5500 $.
- **Part 4**: $ A(0.08) \approx 12208 $.

Would you like further details or have any questions? Here are some related questions to consider:

1. How would the investment amount change if the interest rate were compounded quarterly instead of monthly?
2. How would the IVT be applied if the goal were $8000 instead of $7000?
3. What happens to the function $ A(r) $ as $ r $ approaches 0.08 from the left?
4. Can you derive the exact interest rate that will give an account balance of $7000 using algebraic methods?
5. How does changing the initial investment amount affect the necessary interest rate to reach the same goal?

**Tip**: The Intermediate Value Theorem is a powerful tool when dealing with continuous functions, especially when determining the existence of solutions within specific intervals.

Suppose $5500 is invested in a savings account for 10 years (120 months), with an annual interest rate of r, compounded monthly. The amount of money in the account after 10 years is A(r) = 5500(1 + r/12)^120. Use the Intermediate Value Theorem to show there is a value of r in (0, 0.08) that allows you to reach your savings goal of $7000 in 10 years. Find A(0) and A(0.08) to support this.

Learn how to apply the Intermediate Value Theorem to determine the interest rate required to achieve a savings goal of $7000, given a $5500 investment over 10 years, compounded monthly. This problem includes key concepts such as compound interest and continuous functions, suitable for advanced high school or undergraduate students.

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