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$$f(t)=8,000(0.65)^{t}$$
The given function $f$ models the number of coupons a company sent to their customers at the end of each year, where $t$ represents the number of years since the end of 1998, and $0\leq t\leq 5$. If $y=f(t)$ is graphed in the $ty$-plane, which of the following is the best interpretation of the $y$-intercept of the graph in this context?
Difficulty: Hard
A:
The minimum estimated number of coupons the company sent to their customers during the 5 years was 1,428.
B:
The minimum estimated number of coupons the company sent to their customers during the 5 years was 8,000.
C:
The estimated number of coupons the company sent to their customers at the end of 1998 was 1,428.
D:
The estimated number of coupons the company sent to their customers at the end of 1998 was 8,000.
If $u-3=\frac{6}{t-2}$, what is $t$ in terms of $u$?
Difficulty: Hard
A:
$t=\frac{1}{u}$
B:
$t=\frac{2u+9}{u}$
C:
$t=\frac{1}{u-3}$
D:
$t=\frac{2u}{u-3}$
$$-9x^{2}+30x+c=0$$
In the given equation, $c$ is a constant. The equation has exactly one solution. What is the value of $c$?
Difficulty: Hard
A:
3
B:
0
C:
$-25$
D:
$-53$
$$\frac{x^{2}}{\sqrt{x^{2}-c^{2}}}=\frac{c^{2}}{\sqrt{x^{2}-c^{2}}}+39$$
In the given equation, $c$ is a positive constant. Which of the following is one of the solutions to the given equation?
Difficulty: Hard
A:
$-c$
B:
$-c^{2}-39^{2}$
C:
$-\sqrt{39^{2}-c^{2}}$
D:
$-\sqrt{c^{2}+39^{2}}$
$f(x)=2(3^{x})$
For the function $f$ defined above, what is the value of $f(2)$?
Difficulty: Easy
A:
$9$
B:
$12$
C:
$18$
D:
$36$
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