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題目

解題步驟

1. Understand the question

   a quantity with an initial value of 830 grows exponentially at a rate such that the quantity doubles every 2 weeks. what is the value of the quantity after 21 days, to the nearest hundredth?

2. Explanation

   Step1: Convert time to consistent units

   21 days = $\frac{21}{7}$ = 3 weeks

   Step2: Define exponential growth formula

   The formula for exponential growth with doubling time is $A = A_0 \times 2^{\frac{t}{d}}$, where $A_0$ is initial value, $t$ is time elapsed, $d$ is doubling time.
   Here, $A_0 = 830$, $t=3$, $d=2$.

   Step3: Substitute values into formula

   $A = 830 \times 2^{\frac{3}{2}}$

   Step4: Calculate the exponent and result

   $2^{\frac{3}{2}} = 2\sqrt{2} \approx 2.8284$
   $A \approx 830 \times 2.8284$
   $A \approx 2347.572$

   Step5: Round to nearest hundredth

   Round 2347.572 to two decimal places.

3. Final answer

   2347.57

答案

Question Analysis

OCR Text

a quantity with an initial value of 830 grows exponentially at a rate such that the quantity doubles every 2 weeks. what is the value of the quantity after 21 days, to the nearest hundredth?

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