Math / Algebra

Question $4 a b 6(2,7,35,4$
The formula $A=21.064 e^{00109}$ approximates the populations of Texas, $A$ , in millions, $t$ years after 2000. a. What was the population of Texas in 2000 ? Label units.
What will the population of Texas be in 2025? Round to the nearest million and label units.

Studdy Solution

STEP 1

1. The formula given is $A = 21.064 e^{0.0109t}$ , where $A$ is the population in millions and $t$ is the number of years after 2000.
2. We need to find the population of Texas in the year 2000.
3. We need to find the population of Texas in the year 2025.

STEP 2

1. Determine the population of Texas in 2000 by substituting $t = 0$ into the formula.
2. Calculate the population of Texas in 2025 by substituting $t = 25$ into the formula and rounding the result.

STEP 3

Substitute $t = 0$ into the formula to find the population in 2000:
$A = 21.064 e^{0.0109 \times 0}$

STEP 4

Simplify the expression:
$A = 21.064 e^0$ $A = 21.064 \times 1$ $A = 21.064$
The population of Texas in 2000 was $21.064$ million.

STEP 5

Substitute $t = 25$ into the formula to find the population in 2025:
$A = 21.064 e^{0.0109 \times 25}$

STEP 6

Calculate the exponent:
$0.0109 \times 25 = 0.2725$

STEP 7

Substitute the calculated exponent back into the equation:
$A = 21.064 e^{0.2725}$

STEP 8

Calculate $e^{0.2725}$ :
$e^{0.2725} \approx 1.313$

STEP 9

Substitute the value of $e^{0.2725}$ back into the equation and calculate $A$ :
$A = 21.064 \times 1.313$ $A \approx 27.65$

STEP 10

Round the result to the nearest million:
$A \approx 28$
The population of Texas in 2025 will be approximately $28$ million.
The population of Texas in 2000 was $21.064$ million, and the population in 2025 will be approximately $28$ million.

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Question $4 a b 6(2,7,35,4$
The formula $A=21.064 e^{00109}$ approximates the populations of Texas, $A$ , in millions, $t$ years after 2000. a. What was the population of Texas in 2000 ? Label units.
What will the population of Texas be in 2025? Round to the nearest million and label units.

Studdy Solution

STEP 1

1. The formula given is $A = 21.064 e^{0.0109t}$ , where $A$ is the population in millions and $t$ is the number of years after 2000.
2. We need to find the population of Texas in the year 2000.
3. We need to find the population of Texas in the year 2025.

STEP 2

1. Determine the population of Texas in 2000 by substituting $t = 0$ into the formula.
2. Calculate the population of Texas in 2025 by substituting $t = 25$ into the formula and rounding the result.

STEP 3

Substitute $t = 0$ into the formula to find the population in 2000:
$A = 21.064 e^{0.0109 \times 0}$

STEP 4

Simplify the expression:
$A = 21.064 e^0$ $A = 21.064 \times 1$ $A = 21.064$
The population of Texas in 2000 was $21.064$ million.

STEP 5

Substitute $t = 25$ into the formula to find the population in 2025:
$A = 21.064 e^{0.0109 \times 25}$

STEP 6

Calculate the exponent:
$0.0109 \times 25 = 0.2725$

STEP 7

Substitute the calculated exponent back into the equation:
$A = 21.064 e^{0.2725}$

STEP 8

Calculate $e^{0.2725}$ :
$e^{0.2725} \approx 1.313$

STEP 9

Substitute the value of $e^{0.2725}$ back into the equation and calculate $A$ :
$A = 21.064 \times 1.313$ $A \approx 27.65$

STEP 10

Round the result to the nearest million:
$A \approx 28$
The population of Texas in 2025 will be approximately $28$ million.
The population of Texas in 2000 was $21.064$ million, and the population in 2025 will be approximately $28$ million.

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Studdy Solution

STEP 1

1. The formula given is A=21.064e0.0109t A = 21.064 e^{0.0109t} A=21.064e0.0109t, where A A A is the population in millions and t t t is the number of years after 2000.2. We need to find the population of Texas in the year 2000.3. We need to find the population of Texas in the year 2025.

STEP 2

1. Determine the population of Texas in 2000 by substituting t=0 t = 0 t=0 into the formula.2. Calculate the population of Texas in 2025 by substituting t=25 t = 25 t=25 into the formula and rounding the result.

STEP 3

Substitute t=0 t = 0 t=0 into the formula to find the population in 2000:A=21.064e0.0109×0 A = 21.064 e^{0.0109 \times 0} A=21.064e0.0109×0

STEP 4

Simplify the expression:A=21.064e0 A = 21.064 e^0 A=21.064e0 A=21.064×1 A = 21.064 \times 1 A=21.064×1 A=21.064 A = 21.064 A=21.064The population of Texas in 2000 was 21.064 21.064 21.064 million.

STEP 5

Substitute t=25 t = 25 t=25 into the formula to find the population in 2025:A=21.064e0.0109×25 A = 21.064 e^{0.0109 \times 25} A=21.064e0.0109×25

STEP 6

Calculate the exponent:0.0109×25=0.2725 0.0109 \times 25 = 0.2725 0.0109×25=0.2725

STEP 7

Substitute the calculated exponent back into the equation:A=21.064e0.2725 A = 21.064 e^{0.2725} A=21.064e0.2725

STEP 8

Calculate e0.2725 e^{0.2725} e0.2725:e0.2725≈1.313 e^{0.2725} \approx 1.313 e0.2725≈1.313

STEP 9

Substitute the value of e0.2725 e^{0.2725} e0.2725 back into the equation and calculate A A A:A=21.064×1.313 A = 21.064 \times 1.313 A=21.064×1.313 A≈27.65 A \approx 27.65 A≈27.65

STEP 10

Round the result to the nearest million:A≈28 A \approx 28 A≈28The population of Texas in 2025 will be approximately 28 28 28 million. The population of Texas in 2000 was 21.064 21.064 21.064 million, and the population in 2025 will be approximately 28 28 28 million.

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1. The formula given is $A = 21.064 e^{0.0109t}$ , where $A$ is the population in millions and $t$ is the number of years after 2000.
2. We need to find the population of Texas in the year 2000.
3. We need to find the population of Texas in the year 2025.

1. Determine the population of Texas in 2000 by substituting $t = 0$ into the formula.
2. Calculate the population of Texas in 2025 by substituting $t = 25$ into the formula and rounding the result.

Substitute $t = 0$ into the formula to find the population in 2000:
$A = 21.064 e^{0.0109 \times 0}$

Simplify the expression:
$A = 21.064 e^0$ $A = 21.064 \times 1$ $A = 21.064$
The population of Texas in 2000 was $21.064$ million.

Substitute $t = 25$ into the formula to find the population in 2025:
$A = 21.064 e^{0.0109 \times 25}$

Calculate the exponent:
$0.0109 \times 25 = 0.2725$

Substitute the calculated exponent back into the equation:
$A = 21.064 e^{0.2725}$

Calculate $e^{0.2725}$ :
$e^{0.2725} \approx 1.313$

Substitute the value of $e^{0.2725}$ back into the equation and calculate $A$ :
$A = 21.064 \times 1.313$ $A \approx 27.65$

Round the result to the nearest million:
$A \approx 28$
The population of Texas in 2025 will be approximately $28$ million.
The population of Texas in 2000 was $21.064$ million, and the population in 2025 will be approximately $28$ million.