QuestionAugust 24, 2025

Find the value of k that makes the expression 2500(1+(.075)/(4))^4t equal to the expression (1+(.075)/(4))^4t+k (Round to nearest integer) square

Find the value of k that makes the expression 2500(1+(.075)/(4))^4t equal to the expression (1+(.075)/(4))^4t+k (Round to nearest integer) square
Find the value of k that makes the expression 2500(1+(.075)/(4))^4t equal to the
expression (1+(.075)/(4))^4t+k (Round to nearest integer)
square

Solution
4.0(320 votes)

Answer

373 Explanation 1. Equate the expressions Set 2500(1+\frac{.075}{4})^{4t} = (1+\frac{.075}{4})^{4t+k}. 2. Simplify the equation Divide both sides by (1+\frac{.075}{4})^{4t} to isolate 2500: 2500 = (1+\frac{.075}{4})^k. 3. Solve for k using logarithms Take the natural logarithm of both sides: \ln(2500) = k \cdot \ln(1+\frac{.075}{4}). Solve for k: k = \frac{\ln(2500)}{\ln(1+\frac{.075}{4})}. 4. Calculate the value of k Compute k using a calculator: k \approx \frac{\ln(2500)}{\ln(1.01875)} \approx 372.98. 5. Round to nearest integer Round 372.98 to the nearest integer, which is 373.

Explanation

1. Equate the expressions<br /> Set $2500(1+\frac{.075}{4})^{4t} = (1+\frac{.075}{4})^{4t+k}$.<br /><br />2. Simplify the equation<br /> Divide both sides by $(1+\frac{.075}{4})^{4t}$ to isolate $2500$: <br /> $2500 = (1+\frac{.075}{4})^k$.<br /><br />3. Solve for k using logarithms<br /> Take the natural logarithm of both sides: $\ln(2500) = k \cdot \ln(1+\frac{.075}{4})$.<br /> Solve for $k$: $k = \frac{\ln(2500)}{\ln(1+\frac{.075}{4})}$.<br /><br />4. Calculate the value of k<br /> Compute $k$ using a calculator: $k \approx \frac{\ln(2500)}{\ln(1.01875)} \approx 372.98$.<br /><br />5. Round to nearest integer<br /> Round $372.98$ to the nearest integer, which is $373$.
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