Multiple choice

An account earns $7$ percent interest that is compounded annually. If $ $400$ were invested in the account for $x$ years, find the smallest possible integral value of $x$ such that at the end of $x$ years, the amount in the account will be at least $3$ times the initial investment.

  1. $14$
  2. $15$
  3. $16$
  4. $17$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The amount A = P(1 + r)^x. We want A >= 3P, so (1.07)^x >= 3. Using logarithms, x * log(1.07) >= log(3). x >= log(3) / log(1.07) approx 0.4771 / 0.0294 approx 16.23. The smallest integral value is 17.

AI explanation

Using the compound interest formula Amount equals Principal multiplied by (1 plus Rate divided by 100) raised to the time, we need 400 multiplied by (1.07) to the power of x to be greater than or equal to 1200. This simplifies to (1.07) to the power of x being greater than or equal to 3. Taking the natural logarithm of both sides shows x must be approximately 16.23. Therefore, the smallest possible integral value for x is 17.

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