Solved on Jan 17, 2024

Find the value of $a$ in the equation $g(x)=a \sqrt{a(1-x)}$ if $g(-8)=375$ .

STEP 1

Assumptions
1. The function $g$ is defined as $g(x) = a \sqrt{a(1-x)}$ .
2. We are given that $g(-8) = 375$ .
3. We need to find the value of $a$ .

STEP 2

First, we need to substitute $x = -8$ into the function $g(x)$ to find the equation that will allow us to solve for $a$ .
$g(-8) = a \sqrt{a(1 - (-8))}$

STEP 3

Simplify the equation by evaluating $1 - (-8)$ .
$g(-8) = a \sqrt{a(1 + 8)}$

STEP 4

Further simplify the equation by adding the numbers inside the parentheses.
$g(-8) = a \sqrt{a(9)}$

STEP 5

Now we know that $g(-8) = 375$ , so we can set the equation equal to $375$ .
$375 = a \sqrt{a(9)}$

STEP 6

To solve for $a$ , we need to isolate it. First, we can square both sides of the equation to eliminate the square root.
$(375)^2 = (a \sqrt{a(9)})^2$

STEP 7

Calculate the square of $375$ .
$375^2 = 140625$

STEP 8

Simplify the right side of the equation by squaring the square root, which will leave us with $a \cdot a(9)$ .
$140625 = a^2(9)$

STEP 9

Divide both sides of the equation by $9$ to isolate $a^2$ .
$\frac{140625}{9} = a^2$

STEP 10

Calculate the division on the left side of the equation.
$\frac{140625}{9} = 15625$

STEP 11

Now we have $a^2 = 15625$ . To find $a$ , we take the square root of both sides.
$a = \sqrt{15625}$

STEP 12

Calculate the square root of $15625$ .
$a = 125$
The value of $a$ is $125$ .

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Find the value of $a$ in the equation $g(x)=a \sqrt{a(1-x)}$ if $g(-8)=375$ .

STEP 1

Assumptions
1. The function $g$ is defined as $g(x) = a \sqrt{a(1-x)}$ .
2. We are given that $g(-8) = 375$ .
3. We need to find the value of $a$ .

STEP 2

First, we need to substitute $x = -8$ into the function $g(x)$ to find the equation that will allow us to solve for $a$ .
$g(-8) = a \sqrt{a(1 - (-8))}$

STEP 3

Simplify the equation by evaluating $1 - (-8)$ .
$g(-8) = a \sqrt{a(1 + 8)}$

STEP 4

Further simplify the equation by adding the numbers inside the parentheses.
$g(-8) = a \sqrt{a(9)}$

STEP 5

Now we know that $g(-8) = 375$ , so we can set the equation equal to $375$ .
$375 = a \sqrt{a(9)}$

STEP 6

To solve for $a$ , we need to isolate it. First, we can square both sides of the equation to eliminate the square root.
$(375)^2 = (a \sqrt{a(9)})^2$

STEP 7

Calculate the square of $375$ .
$375^2 = 140625$

STEP 8

Simplify the right side of the equation by squaring the square root, which will leave us with $a \cdot a(9)$ .
$140625 = a^2(9)$

STEP 9

Divide both sides of the equation by $9$ to isolate $a^2$ .
$\frac{140625}{9} = a^2$

STEP 10

Calculate the division on the left side of the equation.
$\frac{140625}{9} = 15625$

STEP 11

Now we have $a^2 = 15625$ . To find $a$ , we take the square root of both sides.
$a = \sqrt{15625}$

STEP 12

Calculate the square root of $15625$ .
$a = 125$
The value of $a$ is $125$ .

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Find the value of aaa in the equation g(x)=aa(1−x)g(x)=a \sqrt{a(1-x)}g(x)=aa(1−x)​ if g(−8)=375g(-8)=375g(−8)=375.

STEP 1

Assumptions1. The function ggg is defined as g(x)=aa(1−x)g(x) = a \sqrt{a(1-x)}g(x)=aa(1−x)​.2. We are given that g(−8)=375g(-8) = 375g(−8)=375.3. We need to find the value of aaa.

STEP 2

First, we need to substitute x=−8x = -8x=−8 into the function g(x)g(x)g(x) to find the equation that will allow us to solve for aaa.g(−8)=aa(1−(−8))g(-8) = a \sqrt{a(1 - (-8))}g(−8)=aa(1−(−8))​

STEP 3

Simplify the equation by evaluating 1−(−8)1 - (-8)1−(−8).g(−8)=aa(1+8)g(-8) = a \sqrt{a(1 + 8)}g(−8)=aa(1+8)​

STEP 4

Further simplify the equation by adding the numbers inside the parentheses.g(−8)=aa(9)g(-8) = a \sqrt{a(9)}g(−8)=aa(9)​

STEP 5

Now we know that g(−8)=375g(-8) = 375g(−8)=375, so we can set the equation equal to 375375375.375=aa(9)375 = a \sqrt{a(9)}375=aa(9)​

STEP 6

To solve for aaa, we need to isolate it. First, we can square both sides of the equation to eliminate the square root.(375)2=(aa(9))2(375)^2 = (a \sqrt{a(9)})^2(375)2=(aa(9)​)2

STEP 7

Calculate the square of 375375375.3752=140625375^2 = 1406253752=140625

STEP 8

Simplify the right side of the equation by squaring the square root, which will leave us with a⋅a(9)a \cdot a(9)a⋅a(9).140625=a2(9)140625 = a^2(9)140625=a2(9)

STEP 9

Divide both sides of the equation by 999 to isolate a2a^2a2.1406259=a2\frac{140625}{9} = a^29140625​=a2

STEP 10

Calculate the division on the left side of the equation.1406259=15625\frac{140625}{9} = 156259140625​=15625

STEP 11

Now we have a2=15625a^2 = 15625a2=15625. To find aaa, we take the square root of both sides.a=15625a = \sqrt{15625}a=15625​

STEP 12

Calculate the square root of 156251562515625.a=125a = 125a=125The value of aaa is 125125125.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the value of $a$ in the equation $g(x)=a \sqrt{a(1-x)}$ if $g(-8)=375$ .

Assumptions
1. The function $g$ is defined as $g(x) = a \sqrt{a(1-x)}$ .
2. We are given that $g(-8) = 375$ .
3. We need to find the value of $a$ .

First, we need to substitute $x = -8$ into the function $g(x)$ to find the equation that will allow us to solve for $a$ .
$g(-8) = a \sqrt{a(1 - (-8))}$

Simplify the equation by evaluating $1 - (-8)$ .
$g(-8) = a \sqrt{a(1 + 8)}$

Further simplify the equation by adding the numbers inside the parentheses.
$g(-8) = a \sqrt{a(9)}$

Now we know that $g(-8) = 375$ , so we can set the equation equal to $375$ .
$375 = a \sqrt{a(9)}$

To solve for $a$ , we need to isolate it. First, we can square both sides of the equation to eliminate the square root.
$(375)^2 = (a \sqrt{a(9)})^2$

Calculate the square of $375$ .
$375^2 = 140625$

Simplify the right side of the equation by squaring the square root, which will leave us with $a \cdot a(9)$ .
$140625 = a^2(9)$

Divide both sides of the equation by $9$ to isolate $a^2$ .
$\frac{140625}{9} = a^2$

Calculate the division on the left side of the equation.
$\frac{140625}{9} = 15625$

Now we have $a^2 = 15625$ . To find $a$ , we take the square root of both sides.
$a = \sqrt{15625}$

Calculate the square root of $15625$ .
$a = 125$
The value of $a$ is $125$ .