Question

to practice her programming skills, kaitlyn wrote a program that takes an input number, squares it, adds a constant value, multiplies the result by another constant value, and then outputs the result. when the input is 8, the output is 204. when the input is 3, the output is 39. what is the output when the input is 5? show your work and solution below.

Explanation:

Step1: Define the function

Let the input number be \( x \), the first constant be \( a \), and the second constant be \( b \). The program's operation can be represented as the function \( f(x)=b(x^{2}+a) \).

Step2: Set up equations

When \( x = 8 \), \( f(8)=b(8^{2}+a)=b(64 + a)=204 \).
When \( x = 3 \), \( f(3)=b(3^{2}+a)=b(9 + a)=39 \).

Step3: Solve the system of equations

We have the system:
\[

$$\begin{cases} b(64 + a)=204 \\ b(9 + a)=39 \end{cases}$$

\]
Divide the first equation by the second equation to eliminate \( b \) (assuming \( b
eq0 \) and \( 9 + a
eq0 \)):
\[
\frac{b(64 + a)}{b(9 + a)}=\frac{204}{39}
\]
Simplify:
\[
\frac{64 + a}{9 + a}=\frac{68}{13}
\]
Cross - multiply:
\( 13(64 + a)=68(9 + a) \)
Expand both sides:
\( 832+13a = 612+68a \)
Subtract \( 13a \) from both sides:
\( 832=612 + 55a \)
Subtract 612 from both sides:
\( 220 = 55a \)
Solve for \( a \):
\( a=\frac{220}{55}=4 \)

Step4: Find the value of \( b \)

Substitute \( a = 4 \) into \( b(9 + a)=39 \):
\( b(9 + 4)=39 \)
\( 13b=39 \)
Solve for \( b \):
\( b=\frac{39}{13}=3 \)

Step5: Find the output for \( x = 5 \)

Now that we know \( a = 4 \) and \( b = 3 \), the function is \( f(x)=3(x^{2}+4) \).
When \( x = 5 \):
\( f(5)=3(5^{2}+4)=3(25 + 4)=3\times29 = 87 \)

Answer:

The output when the input is 5 is \(\boldsymbol{87}\).