Math / Algebra

QuestionA logistic growth model for world population, $\mathrm{f}(\mathrm{x})$ , in billions, x years after 1976 is $\mathrm{f}(\mathrm{x})=\frac{12.57}{1+4.11 e^{-0.026 x}}$ . According to this model, when will the world population be 12 billion?
According to this model, the world population will be 12 billion in $\square$ (Round to the nearest whole number as needed.)

Studdy Solution

STEP 1

1. The logistic growth model is given by the function $\mathrm{f}(\mathrm{x}) = \frac{12.57}{1 + 4.11 e^{-0.026 x}}$ .
2. We need to find the value of $x$ when $\mathrm{f}(\mathrm{x}) = 12$ .
3. The solution will be rounded to the nearest whole number.

STEP 2

1. Set the logistic function equal to 12.
2. Isolate the exponential term.
3. Solve for $x$ .
4. Round the solution to the nearest whole number.

STEP 3

Set the logistic growth function equal to 12:
$\frac{12.57}{1 + 4.11 e^{-0.026 x}} = 12$

STEP 4

Multiply both sides by $1 + 4.11 e^{-0.026 x}$ to eliminate the denominator:
$12.57 = 12(1 + 4.11 e^{-0.026 x})$

STEP 5

Distribute the 12 on the right side:
$12.57 = 12 + 49.32 e^{-0.026 x}$

STEP 6

Subtract 12 from both sides to isolate the exponential term:
$0.57 = 49.32 e^{-0.026 x}$

STEP 7

Divide both sides by 49.32 to solve for the exponential term:
$e^{-0.026 x} = \frac{0.57}{49.32}$

STEP 8

Calculate the right side:
$e^{-0.026 x} \approx 0.011556$

STEP 9

Take the natural logarithm of both sides to solve for $x$ :
$-0.026 x = \ln(0.011556)$

STEP 10

Calculate the natural logarithm:
$-0.026 x \approx -4.4602$

STEP 11

Divide both sides by $-0.026$ to solve for $x$ :
$x \approx \frac{-4.4602}{-0.026}$

STEP 12

Calculate the value of $x$ :
$x \approx 171.54$

STEP 13

Round the value of $x$ to the nearest whole number:
$x \approx 172$
According to this model, the world population will be 12 billion in $\boxed{172}$ years after 1976, which is the year 1976 + 172 = 2148.

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Studdy Solution

STEP 1

1. The logistic growth model is given by the function $\mathrm{f}(\mathrm{x}) = \frac{12.57}{1 + 4.11 e^{-0.026 x}}$ .
2. We need to find the value of $x$ when $\mathrm{f}(\mathrm{x}) = 12$ .
3. The solution will be rounded to the nearest whole number.

STEP 2

1. Set the logistic function equal to 12.
2. Isolate the exponential term.
3. Solve for $x$ .
4. Round the solution to the nearest whole number.

STEP 3

Set the logistic growth function equal to 12:
$\frac{12.57}{1 + 4.11 e^{-0.026 x}} = 12$

STEP 4

Multiply both sides by $1 + 4.11 e^{-0.026 x}$ to eliminate the denominator:
$12.57 = 12(1 + 4.11 e^{-0.026 x})$

STEP 5

Distribute the 12 on the right side:
$12.57 = 12 + 49.32 e^{-0.026 x}$

STEP 6

Subtract 12 from both sides to isolate the exponential term:
$0.57 = 49.32 e^{-0.026 x}$

STEP 7

Divide both sides by 49.32 to solve for the exponential term:
$e^{-0.026 x} = \frac{0.57}{49.32}$

STEP 8

Calculate the right side:
$e^{-0.026 x} \approx 0.011556$

STEP 9

Take the natural logarithm of both sides to solve for $x$ :
$-0.026 x = \ln(0.011556)$

STEP 10

Calculate the natural logarithm:
$-0.026 x \approx -4.4602$

STEP 11

Divide both sides by $-0.026$ to solve for $x$ :
$x \approx \frac{-4.4602}{-0.026}$

STEP 12

Calculate the value of $x$ :
$x \approx 171.54$

STEP 13

Round the value of $x$ to the nearest whole number:
$x \approx 172$
According to this model, the world population will be 12 billion in $\boxed{172}$ years after 1976, which is the year 1976 + 172 = 2148.

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Studdy Solution

STEP 1

STEP 2

1. Set the logistic function equal to 12.2. Isolate the exponential term.3. Solve for x x x.4. Round the solution to the nearest whole number.

STEP 3

Set the logistic growth function equal to 12:12.571+4.11e−0.026x=12 \frac{12.57}{1 + 4.11 e^{-0.026 x}} = 12 1+4.11e−0.026x12.57​=12

STEP 4

Multiply both sides by 1+4.11e−0.026x 1 + 4.11 e^{-0.026 x} 1+4.11e−0.026x to eliminate the denominator:12.57=12(1+4.11e−0.026x) 12.57 = 12(1 + 4.11 e^{-0.026 x}) 12.57=12(1+4.11e−0.026x)

STEP 5

Distribute the 12 on the right side:12.57=12+49.32e−0.026x 12.57 = 12 + 49.32 e^{-0.026 x} 12.57=12+49.32e−0.026x

STEP 6

Subtract 12 from both sides to isolate the exponential term:0.57=49.32e−0.026x 0.57 = 49.32 e^{-0.026 x} 0.57=49.32e−0.026x

STEP 7

Divide both sides by 49.32 to solve for the exponential term:e−0.026x=0.5749.32 e^{-0.026 x} = \frac{0.57}{49.32} e−0.026x=49.320.57​

STEP 8

Calculate the right side:e−0.026x≈0.011556 e^{-0.026 x} \approx 0.011556 e−0.026x≈0.011556

STEP 9

Take the natural logarithm of both sides to solve for x x x:−0.026x=ln⁡(0.011556) -0.026 x = \ln(0.011556) −0.026x=ln(0.011556)

STEP 10

Calculate the natural logarithm:−0.026x≈−4.4602 -0.026 x \approx -4.4602 −0.026x≈−4.4602

STEP 11

Divide both sides by −0.026-0.026−0.026 to solve for x x x:x≈−4.4602−0.026 x \approx \frac{-4.4602}{-0.026} x≈−0.026−4.4602​

STEP 12

Calculate the value of x x x:x≈171.54 x \approx 171.54 x≈171.54

STEP 13

Round the value of x x x to the nearest whole number:x≈172 x \approx 172 x≈172 According to this model, the world population will be 12 billion in 172 \boxed{172} 172​ years after 1976, which is the year 1976 + 172 = 2148.

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1. Set the logistic function equal to 12.
2. Isolate the exponential term.
3. Solve for $x$ .
4. Round the solution to the nearest whole number.

Set the logistic growth function equal to 12:
$\frac{12.57}{1 + 4.11 e^{-0.026 x}} = 12$

Multiply both sides by $1 + 4.11 e^{-0.026 x}$ to eliminate the denominator:
$12.57 = 12(1 + 4.11 e^{-0.026 x})$

Distribute the 12 on the right side:
$12.57 = 12 + 49.32 e^{-0.026 x}$

Subtract 12 from both sides to isolate the exponential term:
$0.57 = 49.32 e^{-0.026 x}$

Divide both sides by 49.32 to solve for the exponential term:
$e^{-0.026 x} = \frac{0.57}{49.32}$

Calculate the right side:
$e^{-0.026 x} \approx 0.011556$

Take the natural logarithm of both sides to solve for $x$ :
$-0.026 x = \ln(0.011556)$

Calculate the natural logarithm:
$-0.026 x \approx -4.4602$

Divide both sides by $-0.026$ to solve for $x$ :
$x \approx \frac{-4.4602}{-0.026}$

Calculate the value of $x$ :
$x \approx 171.54$

Round the value of $x$ to the nearest whole number:
$x \approx 172$
According to this model, the world population will be 12 billion in $\boxed{172}$ years after 1976, which is the year 1976 + 172 = 2148.