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Functions

SAT Functions as Models

         

\[\begin{array}{|c|c|c|c|c|} \hline x & -2 & -1 & 0 & 1\\ \hline g(x) & \frac{9}{4} & \frac{9}{2} & 9 & 18\\ \hline \end{array}\]

The table above shows some values of the function \(g(x)=ba^{x}.\) If \(a\) and \(b\) are nonnegative constants, what is the value of \(a\)?

(A) \(\ \ \frac{1}{4}\)
(B) \(\ \ \frac{1}{2}\)
(C) \(\ \ 1\)
(D) \(\ \ 2\)
(E) \(\ \ 9\)

An experimental population of \(100\) fruit flies increases each hour by \(4\%\). If the approximate number of fruit flies is given by the function \(Q(t)=100x^{t}\), where \(t\) is measured in hours, what is the value of \(x\)?

(A) \(\ \ 0.04\)
(B) \(\ \ 0.6\)
(C) \(\ \ 1.04\)
(D) \(\ \ 1.6\)
(E) \(\ \ 4\)

A stone thrown upward from the top of a building misses the edge of the building on its way down. Its height can be found using the equation

\[y=y_{0}+v_{0}t+\frac{1}{2}at^{2},\]

where \(y_{0}\) is the original position of the stone, \(v_{0}\) is its initial velocity, \(a\) is the acceleration due to gravity, and \(t\) is the time the stone spends in the air. At what time does the stone return to its original height, if \(y_{0}=0\) m, \(v_{0}=25\) m/s, and the acceleration due to gravity is \(-10\) m/s\(^{2}\).

(A) \(\ \ 3\) s
(B) \(\ \ 5\) s
(C) \(\ \ 7\) s
(D) \(\ \ 9\) s
(E) \(\ \ 11\) s

\(^{63}\text{Ni}\), a radioisotope of nickel, has a half-life of \(100\) years. If the quantity of \(^{63}\text{Ni}\) present (in grams) after \(x\) years is given by the function \(y(x)=10\left(\frac{1}{2}\right)^{\frac{x}{100}}\), how much of the initial mass is present after \(80\) years?

(A) \(\ \ 0.80\) g
(B) \(\ \ 1.00\) g
(C) \(\ \ 3.62\) g
(D) \(\ \ 5.74\) g
(E) \(\ \ 80.00\) g

A family-owned juice business has a daily production cost of \(C(x)=5000 - 12x + 2x^{2}\) where \(C\) is the total cost in dollars and \(x\) is the number of juice crates produced. How much greater is the cost of production for \(18\) crates than the cost of production for \(10\) crates?

(A) \(\ \ $352\)
(B) \(\ \ $5032\)
(C) \(\ \ $5432\)
(D) \(\ \ $6232\)
(E) \(\ \ $10512\)

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