, 2024

Since $y = r(x - 3)(x - r)$
passes through $(0,48)$, then $48 = r(0-3)(0-r)$.

Thus, $48 = 3r^2$ and so $r^2 = 16$ or $r
= $\pm$ 4$.
With $13\%$ sales tax on an
item whose price is $\$B$, the total
cost is $\$(1.13B)$.

With $5\%$ sales tax on an item whose
price is $\$B$, the total cost is
$\$(1.05B)$.

From the given information $\$(1.13B) -
\$(1.05B) = \$24$or$1.13B - 1.05B =
24$.

Therefore, $0.08B = 24$, which gives
$B = 300$.

Alternatively, we could note that the difference in total prices is the
difference in the amount of tax paid. This is the difference between
$13\%$ of the original price and
$5\%$ of the original price; this
difference is equal to $8\%$ of the
original price. If $8\%$ of the
original price is equal to $\$24$,
then $1\%$ of the original price is
equal to $\$3$ and so the original
price is $\$3 $\times$ 100 =
\$300$.
When $n=1$, $f(2n) = (f(n))^2$ becomes $f(2) = (f(1))^2$.

Since $f(1) = 3$, then $f(2) = 3^2 = 9$.

When $m=1$, $f(2m+1) = 3f(2m)$ becomes $f(3) = 3f(2)$.

Since $f(2) = 9$, then $f(3) = 3 \cdot 9 = 27$.

When $n=2$, $f(2n) = (f(n))^2$ becomes $f(4) = (f(2))^2$.

Since $f(2) = 9$, then $f(4) = 9^2 = 81$.

Therefore, $f(2) + f(3) + f(4) = 9 + 27 + 81
= 117$.