Math / Algebra

QuestionThe formula $A=19.6 e^{0.0139 t}$ models the population of a US state, A , in millions, t years after 2000. a. What was the population of the state in 2000 ? b. When will the population of the state reach 25.9 million? a. In 2000, the population of the state was $\square$ million.

Studdy Solution

STEP 1

1. We are given the formula $A = 19.6 e^{0.0139 t}$ which models the population $A$ in millions, $t$ years after 2000.
2. We need to find the population in the year 2000.
3. We need to determine when the population will reach 25.9 million.

STEP 2

1. Calculate the population in the year 2000.
2. Set up an equation to find when the population will reach 25.9 million.
3. Solve the equation for $t$ .

STEP 3

To find the population in 2000, substitute $t = 0$ into the formula:
$A = 19.6 e^{0.0139 \times 0}$

STEP 4

Simplify the expression:
$A = 19.6 e^0$ $A = 19.6 \times 1$ $A = 19.6$
The population in the year 2000 was 19.6 million.

STEP 5

Set up the equation for when the population reaches 25.9 million:
$25.9 = 19.6 e^{0.0139 t}$

STEP 6

Solve for $t$ . First, divide both sides by 19.6:
$\frac{25.9}{19.6} = e^{0.0139 t}$

STEP 7

Take the natural logarithm of both sides to solve for $t$ :
$\ln\left(\frac{25.9}{19.6}\right) = 0.0139 t$

STEP 8

Solve for $t$ by dividing both sides by 0.0139:
$t = \frac{\ln\left(\frac{25.9}{19.6}\right)}{0.0139}$
The population of the state in 2000 was $\boxed{19.6}$ million.
To find when the population will reach 25.9 million, calculate $t$ using the equation from STEP_6.

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QuestionThe formula $A=19.6 e^{0.0139 t}$ models the population of a US state, A , in millions, t years after 2000. a. What was the population of the state in 2000 ? b. When will the population of the state reach 25.9 million? a. In 2000, the population of the state was $\square$ million.

Studdy Solution

STEP 1

1. We are given the formula $A = 19.6 e^{0.0139 t}$ which models the population $A$ in millions, $t$ years after 2000.
2. We need to find the population in the year 2000.
3. We need to determine when the population will reach 25.9 million.

STEP 2

1. Calculate the population in the year 2000.
2. Set up an equation to find when the population will reach 25.9 million.
3. Solve the equation for $t$ .

STEP 3

To find the population in 2000, substitute $t = 0$ into the formula:
$A = 19.6 e^{0.0139 \times 0}$

STEP 4

Simplify the expression:
$A = 19.6 e^0$ $A = 19.6 \times 1$ $A = 19.6$
The population in the year 2000 was 19.6 million.

STEP 5

Set up the equation for when the population reaches 25.9 million:
$25.9 = 19.6 e^{0.0139 t}$

STEP 6

Solve for $t$ . First, divide both sides by 19.6:
$\frac{25.9}{19.6} = e^{0.0139 t}$

STEP 7

Take the natural logarithm of both sides to solve for $t$ :
$\ln\left(\frac{25.9}{19.6}\right) = 0.0139 t$

STEP 8

Solve for $t$ by dividing both sides by 0.0139:
$t = \frac{\ln\left(\frac{25.9}{19.6}\right)}{0.0139}$
The population of the state in 2000 was $\boxed{19.6}$ million.
To find when the population will reach 25.9 million, calculate $t$ using the equation from STEP_6.

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Studdy Solution

STEP 1

1. We are given the formula A=19.6e0.0139t A = 19.6 e^{0.0139 t} A=19.6e0.0139t which models the population A A A in millions, t t t years after 2000.2. We need to find the population in the year 2000.3. We need to determine when the population will reach 25.9 million.

STEP 2

1. Calculate the population in the year 2000.2. Set up an equation to find when the population will reach 25.9 million.3. Solve the equation for t t t.

STEP 3

To find the population in 2000, substitute t=0 t = 0 t=0 into the formula:A=19.6e0.0139×0 A = 19.6 e^{0.0139 \times 0} A=19.6e0.0139×0

STEP 4

Simplify the expression:A=19.6e0 A = 19.6 e^0 A=19.6e0 A=19.6×1 A = 19.6 \times 1 A=19.6×1 A=19.6 A = 19.6 A=19.6The population in the year 2000 was 19.6 million.

STEP 5

Set up the equation for when the population reaches 25.9 million:25.9=19.6e0.0139t 25.9 = 19.6 e^{0.0139 t} 25.9=19.6e0.0139t

STEP 6

Solve for t t t. First, divide both sides by 19.6:25.919.6=e0.0139t \frac{25.9}{19.6} = e^{0.0139 t} 19.625.9​=e0.0139t

STEP 7

Take the natural logarithm of both sides to solve for t t t:ln⁡(25.919.6)=0.0139t \ln\left(\frac{25.9}{19.6}\right) = 0.0139 t ln(19.625.9​)=0.0139t

STEP 8

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1. We are given the formula $A = 19.6 e^{0.0139 t}$ which models the population $A$ in millions, $t$ years after 2000.
2. We need to find the population in the year 2000.
3. We need to determine when the population will reach 25.9 million.

1. Calculate the population in the year 2000.
2. Set up an equation to find when the population will reach 25.9 million.
3. Solve the equation for $t$ .

To find the population in 2000, substitute $t = 0$ into the formula:
$A = 19.6 e^{0.0139 \times 0}$

Simplify the expression:
$A = 19.6 e^0$ $A = 19.6 \times 1$ $A = 19.6$
The population in the year 2000 was 19.6 million.

Set up the equation for when the population reaches 25.9 million:
$25.9 = 19.6 e^{0.0139 t}$

Solve for $t$ . First, divide both sides by 19.6:
$\frac{25.9}{19.6} = e^{0.0139 t}$

Take the natural logarithm of both sides to solve for $t$ :
$\ln\left(\frac{25.9}{19.6}\right) = 0.0139 t$