Questions: Suppose the value of an investment doubles every 7 years. By what factor will its value rise in 30 years? (A) 2^(30 / 7) (B) 2^30-2^7 (C) 7 × 2^30

Suppose the value of an investment doubles every 7 years. By what factor will its value rise in 30 years? (A) 2^(30 / 7) (B) 2^30-2^7 (C) 7 × 2^30
Transcript text: Suppose the value of an investment doubles every 7 years. By what factor will its value rise in 30 years? (A) $2^{30 / 7}$ (B) $2^{30}-2^{7}$ (C) $7 \times 2^{30}$
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Solution

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Solution Steps

To determine the factor by which the investment's value will rise in 30 years, we need to calculate how many times the investment doubles in that period. Since it doubles every 7 years, we divide 30 by 7 to find the number of doubling periods. The factor is then \(2\) raised to the power of the number of doubling periods.

Step 1: Determine the Number of Doubling Periods

The investment doubles every 7 years. To find the number of doubling periods in 30 years, we calculate: \[ \text{doubling\_periods} = \frac{30}{7} \approx 4.2857 \]

Step 2: Calculate the Factor of Increase

The factor by which the investment's value rises after 30 years is given by: \[ \text{factor} = 2^{\text{doubling\_periods}} = 2^{4.2857} \approx 19.5042 \]

Final Answer

The factor by which the investment's value will rise in 30 years is approximately \(19.5042\). Thus, the answer is: \[ \boxed{19.5042} \]

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