If `g(x)=asqrt(a(1-x))` Function g is defined by the equation above. If `g(-8)=375`, what is the value of a?

Options

A. `25`
B. `75`
C. `125`
D. `625`

Answer

To solve the problem, we start with the function given:

\[ g(x) = a \sqrt{a(1-x)} \]

We know that:

\[ g(-8) = 375 \]

Substituting \(-8\) into the function, we have:

\[ g(-8) = a \sqrt{a(1 - (-8))} = a \sqrt{a(1 + 8)} = a \sqrt{a \cdot 9} = a \sqrt{9a} \]

Since \(\sqrt{9} = 3\), we can simplify this to:

\[ g(-8) = a \cdot 3 \sqrt{a} = 3a \sqrt{a} \]

Setting this equal to 375, we have:

\[ 3a \sqrt{a} = 375 \]

Now, we can divide both sides by 3:

\[ a \sqrt{a} = \frac{375}{3} = 125 \]

Next, we can express \(\sqrt{a}\) as \(a^{1/2}\):

\[ a \cdot a^{1/2} = a^{1 + 1/2} = a^{3/2} \]

Thus, we have:

\[ a^{3/2} = 125 \]

We can rewrite 125 as \(5^3\):

\[ a^{3/2} = 5^3 \]

To solve for \(a\), we will raise both sides to the power of \(\frac{2}{3}\):

\[ a = (5^3)^{\frac{2}{3}} = 5^{3 \cdot \frac{2}{3}} = 5^2 = 25 \]

Thus, the value of \(a\) is:

\[ \boxed{25} \]