Assume you invest 250 at the end of each year for

Questions: Assume you invest 250 at the end of each year for 10 years at an annual interest rate of r. The amount of money in the account after 10 years is (A=frac250[(1+r)^10-1]r), Your goal is to have 3500 in your account after 10 years (a) Let (f) be the function such that (f(x)=frac250[(1+r)^10-1]r), Using the Intermediate Value Theorem, determine whether there is an interest rate (r) in (0.01,0.10), between (1 %) and (10 %), that allows you to reach your financial goal. Choose the correct answer below. A. No, the value of (r) is greater than (10 %) B. Yes, because (f(0.01)<250<f(0.10)). C. No, it is not possible to reach the goal of 3500 D. Yes, because the function is continuous on ([0.01,0.10]) and 3500 is between (f(0.01)) and (f(0.10)) (b) Use a calculator to estimate the interest rate required to reach your financial goal.

$Assume you invest 250 at the end of each year for 10 years at an annual interest rate of r. The amount of money in the account after 10 years is (A=frac250[(1+r)^10-1]r), Your goal is to have 3500 in your account after 10 years (a) Let (f) be the function such that (f(x)=frac250[(1+r)^10-1]r), Using the Intermediate Value Theorem, determine whether there is an interest rate (r) in (0.01,0.10), between (1 %) and (10 %), that allows you to reach your financial goal. Choose the correct answer below. A. No, the value of (r) is greater than (10 %) B. Yes, because (f(0.01)<250<f(0.10)). C. No, it is not possible to reach the goal of 3500 D. Yes, because the function is continuous on ([0.01,0.10]) and 3500 is between (f(0.01)) and (f(0.10)) (b) Use a calculator to estimate the interest rate required to reach your financial goal.$

Transcript text: Assume you invest $\$ 250$ at the end of each year for 10 years at an annual interest rate of r . The amount of money in the account after 10 years is $A=\frac{\left.250 \mid[1+r)^{10}-1\right]}{r}$, Your goal is to have $\$ 3500$ in your account after 10 years (a) Let $f$ be the function such that $f(x)=\frac{250\left[(1+r)^{10}-1\right]}{r}$, Using the Intermediate Value Theorem, determine whether there is an interest rate $r$ in $(0.01,0.10)$, between $1 \%$ and $10 \%$, that allows you to reach your financial goal. Choose the correct answer below. A. No, the value of $r$ is greater than $10 \%$ B. Yes, because $f(0.01)<250

Solution

Solution Steps

To solve this problem, we need to evaluate the function $ f(r) = \frac{250[(1+r)^{10}-1]}{r} $ at the endpoints of the interval $ (0.01, 0.10) $ and check if the value 3500 lies between $ f(0.01) $ and $ f(0.10) $. This will help us determine if there is an interest rate $ r $ in the given interval that allows us to reach the financial goal using the Intermediate Value Theorem. Then, we can use a numerical method to estimate the interest rate required to reach the financial goal.

Solution Approach

Define the function $ f(r) $.
Evaluate $ f(0.01) $ and $ f(0.10) $.
Check if 3500 lies between $ f(0.01) $ and $ f(0.10) $.
Use a numerical method (e.g., binary search or a root-finding algorithm) to estimate the interest rate $ r $ that makes $ f(r) = 3500 $.

Step 1: Define the Function $ f(r) $

The function $ f(r) $ is given by: \[ f(r) = \frac{250 \left[(1+r)^{10} - 1\right]}{r} \]

Step 2: Evaluate the Function at the Endpoints

Evaluate $ f(r) $ at $ r = 0.01 $ and $ r = 0.10 $: \[ f(0.01) = 2615.5531 \] \[ f(0.10) = 3984.3562 \]

Step 3: Apply the Intermediate Value Theorem

Since $ f(r) $ is continuous on $[0.01, 0.10]$ and $ 3500 $ lies between $ f(0.01) $ and $ f(0.10) $, by the Intermediate Value Theorem, there exists an $ r $ in $(0.01, 0.10)$ such that $ f(r) = 3500 $.

Step 4: Estimate the Interest Rate

Using a numerical method, we estimate the interest rate $ r $ that satisfies $ f(r) = 3500 $: \[ r \approx 0.0728 \]