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Question Quantity with initial value of 8900 grows exponentially at 65%65\% per hour. Find the value after 411 minutes, rounded to nearest hundredth.

Studdy Solution

STEP 1

Assumptions
1. The initial value of the quantity is 8900.
2. The growth rate is 65% per hour.
3. The time for which we need to find the final value is 411 minutes.
4. The growth is exponential.
5. We need to find the final value to the nearest hundredth.

STEP 2

First, we need to convert the growth rate from a percentage to a decimal to use it in calculations.
65%=0.6565\% = 0.65

STEP 3

Since the growth rate is given per hour, we need to convert the time from minutes to hours because the time given is in minutes.
Time in hours=Time in minutesMinutes per hourTime\ in\ hours = \frac{Time\ in\ minutes}{Minutes\ per\ hour}

STEP 4

Plug in the values for the time in minutes and minutes per hour to calculate the time in hours.
Time in hours=41160Time\ in\ hours = \frac{411}{60}

STEP 5

Calculate the time in hours.
Time in hours=41160≈6.85Time\ in\ hours = \frac{411}{60} \approx 6.85

STEP 6

The formula for exponential growth is given by:
Final value=Initial value×(1+Growth rate)TimeFinal\ value = Initial\ value \times (1 + Growth\ rate)^{Time}

STEP 7

Now, plug in the values for the initial value, growth rate, and time in hours into the exponential growth formula.
Final value=8900×(1+0.65)6.85Final\ value = 8900 \times (1 + 0.65)^{6.85}

STEP 8

Calculate the value inside the parentheses.
1+0.65=1.651 + 0.65 = 1.65

STEP 9

Now, plug this value into the formula.
Final value=8900×1.656.85Final\ value = 8900 \times 1.65^{6.85}

STEP 10

Calculate the final value using the exponential function.
Final value≈8900×1.656.85Final\ value \approx 8900 \times 1.65^{6.85}

STEP 11

Use a calculator to compute the final value to the nearest hundredth.
Final value≈8900×1.656.85≈8900×58.211Final\ value \approx 8900 \times 1.65^{6.85} \approx 8900 \times 58.211

STEP 12

Multiply the initial value by the calculated exponential factor.
Final value≈8900×58.211≈518078.9Final\ value \approx 8900 \times 58.211 \approx 518078.9

STEP 13

Round the final value to the nearest hundredth.
Final value≈518078.90Final\ value \approx 518078.90
The value of the quantity after 411 minutes, to the nearest hundredth, is approximately $518078.90.

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