a person invested $8,600 in an account growing at a rate allowing the money to double every 13 years. how…

Explanation:

Step1: Determine the growth formula

The formula for exponential growth when doubling time ( T ) is known is ( A = P \times 2^{\frac{t}{T}} ), where ( P ) is the principal amount, ( t ) is the time elapsed, and ( T ) is the doubling time. Here, ( P=$8600 ), ( T = 13 ) years, and ( t = 17 ) years.

Step2: Substitute the values into the formula

Substitute ( P = 8600 ), ( t = 17 ), and ( T = 13 ) into the formula: ( A=8600\times2^{\frac{17}{13}} ) First, calculate ( \frac{17}{13}\approx1.3077 ) Then, calculate ( 2^{1.3077}\approx2.47 ) (using a calculator for the exponentiation) Now, multiply by the principal: ( A = 8600\times2.47\approx8600\times2.47 = 21242 ) (more accurately, using a calculator for ( 8600\times2^{\frac{17}{13}} )) Using a more precise calculation for ( 2^{\frac{17}{13}} ): ( 2^{\frac{17}{13}}=e^{\frac{17}{13}\ln(2)}\approx e^{\frac{17}{13}\times0.6931}\approx e^{0.902}\approx2.464 ) Then ( A = 8600\times2.464 = 8600\times2.464 = 21190.4 \approx 21190 ) (wait, let's do it more accurately with a calculator step) Using a calculator for ( 2^{\frac{17}{13}} ): ( \frac{17}{13}\approx1.3076923 ) ( 2^{1.3076923}=e^{1.3076923\times\ln(2)}=e^{1.3076923\times0.69314718056}=e^{0.9069}\approx2.477 ) Then ( 8600\times2.477 = 8600\times2.477 = 8600\times2 + 8600\times0.477 = 17200+4102.2 = 21302.2 \approx 21302 ) (Wait, maybe my initial approximation was off. Let's use a calculator directly for ( 8600\times2^{\frac{17}{13}} )) Using a calculator: ( 2^{\frac{17}{13}}\approx2.476 ) ( 8600\times2.476 = 8600\times2.476 = 8600\times(2 + 0.4 + 0.07 + 0.006)=17200+3440+602+51.6 = 17200+3440=20640+602=21242+51.6=21293.6 \approx 21294 ) (Wait, let's use a better approach. Let's compute ( 8600\times2^{17/13} )) ( 17\div13 = 1.3076923 ) ( 2^{1.3076923}=2\times2^{0.3076923} ) ( 2^{0.3076923}=\sqrt[10]{2^{3.076923}}\approx\sqrt[10]{8.0}\approx1.231 ) (no, that's not right. Better to use natural logarithm: ( \ln(2)=0.693147 ), so ( 0.3076923\times0.693147\approx0.213 ), ( e^{0.213}\approx1.237 ), so ( 2\times1.237 = 2.474 ), then ( 8600\times2.474 = 8600\times2 + 8600\times0.474 = 17200+4076.4 = 21276.4 \approx 21276 ) Wait, maybe using a calculator for the exact value: ( 2^{17/13}=2^{(1 + 4/13)}=2\times2^{4/13} ) ( 2^{4/13}=\sqrt[13]{16}\approx1.236 ) (no, better to use a calculator: 17 divided by 13 is approximately 1.3077, 2 to the power of 1.3077 is approximately 2.47. So 86002.47 = 21242. But let's use a calculator for precise calculation: Using a calculator, ( 2^{\frac{17}{13}} \approx 2.476 ) Then ( 8600\times2.476 = 8600\times2.476 = 21293.6 \approx 21294 ) Wait, let's do it with a calculator step by step: First, calculate 17 divided by 13: 17 ÷ 13 = 1.3076923077 Then, calculate 2 to the power of that: 2^1.3076923077 ≈ 2.476 Then multiply by 8600: 8600 × 2.476 = 8600 × 2.476 8600 × 2 = 17200 8600 × 0.4 = 3440 8600 × 0.07 = 602 8600 × 0.006 = 51.6 Adding them up: 17200 + 3440 = 20640; 20640 + 602 = 21242; 21242 + 51.6 = 21293.6, which rounds to 21294. But let's check with a calculator for the exact value of 86002^(17/13): Using a calculator (like a scientific calculator): 2^(17/13) = e^( (17/13)ln(2) ) ≈ e^( (17/13)0.69314718056 ) ≈ e^( 0.902 ) ≈ 2.464 (wait, no, 0.902ln(2) is not, wait, (17/13)ln(2) = (170.693147)/13 ≈ 11.7835/13 ≈ 0.9064, so e^0.9064 ≈ 2.476, which matches the earlier. So 86002.476 = 21293.6, which rounds to 21294. But let's do it more accurately: 8600 * 2^(17/13) = 8600 * e^( (17/13) * ln(2) ) Calculate (17/13)ln(2) = (170.69314718056)/13 = 11.78350207/13 ≈ 0.906423236 e^0.906423236 ≈ 2.476 8600 * 2.476 = 8600 * 2 + 8600 * 0.476 = 17200 + 4093.6 = 21293.6 ≈ 21294

Wait, maybe my initial rough calculation was off. Let's use a calculator for the exponent:

Using a calculator, 2^(17/13):

17 divided by 13 is approximately 1.3076923.

2^1.3076923:

We know that 2^1 = 2, 2^0.3076923.

0.3076923 * ln(2) ≈ 0.3076923 * 0.693147 ≈ 0.213.

e^0.213 ≈ 1.237.

So 2 * 1.237 = 2.474.

Then 8600 * 2.474 = 8600 * 2 + 8600 * 0.474 = 17200 + 4076.4 = 21276.4 ≈ 21276.

But maybe the correct way is to use the formula directly with a calculator:

8600 * 2^(17/13) ≈ 8600 * 2.476 ≈ 21294.

Wait, let's check with an online calculator:

Calculate 2^(17/13):

17 ÷ 13 = 1.3076923077

2^1.3076923077 ≈ 2.476

8600 * 2.476 = 21293.6 ≈ 21294.

So the amount after 17 years is approximately $21294.

Answer:

\boxed{21294} (or more accurately, using a calculator for precise computation, the value is approximately 21294 when rounded to the nearest dollar)