(b) How many years does it take the balance to first exceed $$ \$ 1,700 $$ ? Solve with a graph or table and report your answer accurate to the nearest 0.1 years. $$ t \approx $$ Number years (Round to the nearest 0.1 year.)

Question

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We start with the balance function

$$
A(t)=850(1.07)^t,
$$

and we want to know for what $$ t $$ the balance first exceeds $$\$1700$$. This means we must solve

$$
850(1.07)^t = 1700.
$$

Notice that $$1700$$ is twice $$850$$, so we can write

$$
850(1.07)^t = 2\,(850).
$$

Dividing both sides by $$850$$ gives

$$
(1.07)^t = 2.
$$

To solve for $$t$$ we take the natural logarithm of both sides:

$$
\ln\bigl((1.07)^t\bigr) = \ln 2.
$$

Using the logarithm power rule, we get

$$
t\ln(1.07) = \ln 2,
$$

and therefore

$$
t=\frac{\ln 2}{\ln(1.07)}.
$$

Now, using a calculator:

$$
\ln 2\approx0.6931,\quad \ln(1.07)\approx0.06766,
$$

so

$$
t\approx \frac{0.6931}{0.06766}\approx 10.244.
$$

Rounded to the nearest 0.1 year, we have

$$
t\approx10.2\text{ years}.
$$

To confirm this result, we can construct a small table that shows approximate values of $$A(t)$$:

| $$ t $$ (years) | Calculation                           | $$ A(t) $$ (approx.) |
|-----------------|---------------------------------------|----------------------|
| 10.0            | $$850(1.07)^{10.0}$$  ≈ $$850(1.967)$$  | \$1672               |
| 10.2            | $$850(1.07)^{10.2}$$  ≈ $$850(1.995)$$  | \$1696               |
| 10.3            | $$850(1.07)^{10.3}$$  ≈ $$850(2.007)$$  | \$1705               |

From the table, we see the balance is below $$\$1700$$ at $$t=10.2$$ years and exceeds $$\$1700$$ by $$t=10.3$$ years, which agrees with our exact computation showing the threshold near $$10.244$$ years. Rounding to the nearest $$0.1$$ year gives

$$
\boxed{10.2 \text{ years}}.
$$
It takes approximately 10.2 years for the balance to first exceed \$1,700.

(b) How many years does it take the balance to first exceed \( \$ 1,700 \) ? Solve with a graph or table and report your answer accurate to the nearest 0.1 years. \( t \approx \) Number years (Round to the nearest 0.1 year.)

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