Find the interest rate (with annual compounding) that makes the statement true. Round to the nearest tenth when
necessary.
grows to in 2 years

Ask by Bates Johnston. in the United States

Mar 20,2025

Question

Find the interest rate (with annual compounding) that makes the statement true. Round to the nearest tenth when necessary. $$ \$ 10,000 $$ grows to $$ \$ 11,236.00 $$ in 2 years

UpStudy AI · Accepted Answer

To find the interest rate that makes the statement true, we can use the formula for compound interest:

$$
A = P(1 + r)^t
$$

Where:
- $$ A $$ is the amount of money accumulated after n years, including interest.
- $$ P $$ is the principal amount (the initial amount of money).
- $$ r $$ is the annual interest rate (decimal).
- $$ t $$ is the time the money is invested or borrowed for, in years.

Given:
- $$ A = 11,236.00 $$
- $$ P = 10,000 $$
- $$ t = 2 $$

We need to solve for $$ r $$. Plugging in the known values, we have:

$$
11,236 = 10,000(1 + r)^2
$$

Now, we can isolate $$ (1 + r)^2 $$:

$$
(1 + r)^2 = \frac{11,236}{10,000}
$$

Calculating the right side:

$$
(1 + r)^2 = 1.1236
$$

Next, we take the square root of both sides to solve for $$ 1 + r $$:

$$
1 + r = \sqrt{1.1236}
$$

Now, we can calculate $$ \sqrt{1.1236} $$ and then isolate $$ r $$:

$$
r = \sqrt{1.1236} - 1
$$

Let's calculate this step by step.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{1.1236}$$
- step1: Rewrite the expression:
  $$\sqrt{\frac{2809}{2500}}$$
- step2: Use the properties of radicals:
  $$\frac{\sqrt{2809}}{\sqrt{2500}}$$
- step3: Simplify the expression:
  $$\frac{53}{50}$$
Calculate or simplify the expression $$ \sqrt(1.1236) - 1 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{1.1236}-1$$
- step1: Simplify the root:
  $$\frac{53}{50}-1$$
- step2: Reduce fractions to a common denominator:
  $$\frac{53}{50}-\frac{50}{50}$$
- step3: Transform the expression:
  $$\frac{53-50}{50}$$
- step4: Subtract the numbers:
  $$\frac{3}{50}$$
Calculating the square root of $$ 1.1236 $$:

$$
\sqrt{1.1236} \approx 1.06
$$

Now, substituting this back to find $$ r $$:

$$
r = 1.06 - 1 = 0.06
$$

To express $$ r $$ as a percentage, we multiply by 100:

$$
r \approx 0.06 \times 100 = 6\%
$$

Thus, the annual interest rate that makes the statement true is approximately:

$$
\boxed{6.0\%}
$$
The interest rate is approximately 6.0%.

Find the interest rate (with annual compounding) that makes the statement true. Round to the nearest tenth when
necessary.
grows to in 2 years

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Find the interest rate (with annual compounding) that makes the statement true. Round to the nearest tenth when necessary. grows to in 2 years