QuestionA child is 20 inches long at birth. Use the function $f(x)=20+47 \log (x+2)$ to find when she reaches $60\%$ of her height.

Studdy Solution

STEP 1

Assumptions1. The child's length at birth is20 inches. . The child's growth rate is proportional to her body size.
3. The percentage of the child's adult height attained can be modeled by the function $f(x)=20+47 \log (x+)$ , where $x$ represents her age in years and $f(x)$ represents the percentage of her adult height reached at age $x$ .

STEP 2

We are asked to find the age at which the child will reach60% of her adult height. This means we need to solve the equation $f(x) =60$ for $x$ .

STEP 3

Substitute $60$ for $f(x)$ in the equation.
$60 =20+47 \log (x+2)$

STEP 4

Subtract $20$ from both sides of the equation to isolate the logarithmic term.
$60 -20 =47 \log (x+2)$

STEP 5

implify the left side of the equation.
$40 =47 \log (x+2)$

STEP 6

Divide both sides of the equation by $47$ to isolate the logarithm.
$\frac{40}{47} = \log (x+2)$

STEP 7

To solve for $x$ , we need to remove the logarithm. We can do this by raising both sides of the equation as powers of $10$ (since the base of the logarithm is $10$ ).
$10^{\frac{40}{47}} = x+2$

STEP 8

Subtract $2$ from both sides of the equation to solve for $x$ .
$10^{\frac{40}{47}} -2 = x$

STEP 9

Calculate the value of $x$ .
$x \approx.5$ Since we are asked to round the answer to the nearest whole year, the child will reach60% of her adult height at approximately2 years of age.

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QuestionA child is 20 inches long at birth. Use the function $f(x)=20+47 \log (x+2)$ to find when she reaches $60\%$ of her height.

Studdy Solution

STEP 1

STEP 2

We are asked to find the age at which the child will reach60% of her adult height. This means we need to solve the equation $f(x) =60$ for $x$ .

STEP 3

Substitute $60$ for $f(x)$ in the equation.
$60 =20+47 \log (x+2)$

STEP 4

Subtract $20$ from both sides of the equation to isolate the logarithmic term.
$60 -20 =47 \log (x+2)$

STEP 5

implify the left side of the equation.
$40 =47 \log (x+2)$

STEP 6

Divide both sides of the equation by $47$ to isolate the logarithm.
$\frac{40}{47} = \log (x+2)$

STEP 7

To solve for $x$ , we need to remove the logarithm. We can do this by raising both sides of the equation as powers of $10$ (since the base of the logarithm is $10$ ).
$10^{\frac{40}{47}} = x+2$

STEP 8

Subtract $2$ from both sides of the equation to solve for $x$ .
$10^{\frac{40}{47}} -2 = x$

STEP 9

Calculate the value of $x$ .
$x \approx.5$ Since we are asked to round the answer to the nearest whole year, the child will reach60% of her adult height at approximately2 years of age.

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QuestionA child is 20 inches long at birth. Use the function f(x)=20+47log⁡(x+2)f(x)=20+47 \log (x+2)f(x)=20+47log(x+2) to find when she reaches 60%60\%60% of her height.

Studdy Solution

STEP 1

STEP 2

We are asked to find the age at which the child will reach60% of her adult height. This means we need to solve the equation f(x)=60f(x) =60f(x)=60 for xxx.

STEP 3

Substitute 606060 for f(x)f(x)f(x) in the equation.60=20+47log⁡(x+2)60 =20+47 \log (x+2)60=20+47log(x+2)

STEP 4

Subtract 202020 from both sides of the equation to isolate the logarithmic term.60−20=47log⁡(x+2)60 -20 =47 \log (x+2)60−20=47log(x+2)

STEP 5

implify the left side of the equation.40=47log⁡(x+2)40 =47 \log (x+2)40=47log(x+2)

STEP 6

Divide both sides of the equation by 474747 to isolate the logarithm.4047=log⁡(x+2)\frac{40}{47} = \log (x+2)4740​=log(x+2)

STEP 7

To solve for xxx, we need to remove the logarithm. We can do this by raising both sides of the equation as powers of 101010 (since the base of the logarithm is 101010).104047=x+210^{\frac{40}{47}} = x+2104740​=x+2

STEP 8

Subtract 222 from both sides of the equation to solve for xxx.104047−2=x10^{\frac{40}{47}} -2 = x104740​−2=x

STEP 9

Calculate the value of xxx.x≈.5x \approx.5x≈.5Since we are asked to round the answer to the nearest whole year, the child will reach60% of her adult height at approximately2 years of age.

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QuestionA child is 20 inches long at birth. Use the function $f(x)=20+47 \log (x+2)$ to find when she reaches $60\%$ of her height.

We are asked to find the age at which the child will reach60% of her adult height. This means we need to solve the equation $f(x) =60$ for $x$ .

Substitute $60$ for $f(x)$ in the equation.
$60 =20+47 \log (x+2)$

Subtract $20$ from both sides of the equation to isolate the logarithmic term.
$60 -20 =47 \log (x+2)$

implify the left side of the equation.
$40 =47 \log (x+2)$

Divide both sides of the equation by $47$ to isolate the logarithm.
$\frac{40}{47} = \log (x+2)$

To solve for $x$ , we need to remove the logarithm. We can do this by raising both sides of the equation as powers of $10$ (since the base of the logarithm is $10$ ).
$10^{\frac{40}{47}} = x+2$

Subtract $2$ from both sides of the equation to solve for $x$ .
$10^{\frac{40}{47}} -2 = x$

Calculate the value of $x$ .
$x \approx.5$ Since we are asked to round the answer to the nearest whole year, the child will reach60% of her adult height at approximately2 years of age.