Practice Test 12 - AP Precalculus

20 questions. Answer them in any order, then submit to see your score and every explanation.

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1. A city models population over decades, but officials note limited housing and water supply. Model 1: Exponential, $y=50000(1.03)^x$, where 1.03 is the decade growth factor. Model 2: Logarithmic, $y=50000+18000\log(x+1)$, where 18000 controls the increase size. Early census counts rise quickly and resemble the exponential curve for two decades. After that, growth slows as new building permits become scarce. The exponential model keeps accelerating and predicts extreme populations by year 80. The logarithmic model still increases but at a decreasing rate, consistent with resource limits. Based on the models described in the passage, what are the implications of choosing an exponential model over a logarithmic model for long-term predictions?
2. What does the absolute value of the determinant of a $2 \times 2$ transformation matrix represent geometrically?
3. A hot drink cools as $T(t)=20+70(0.92)^t$. When will $T(t)=40^\circ\text{C}$?
4. For $h(x)=\dfrac{x+2}{(x-1)(x+3)}$, identify the vertical asymptote(s).
5. A car is worth $V_0=\$24{,}000$. Linear depreciation: $V_L(t)=24{,}000-1{,}800t$, where $t$ is years and $\$1{,}800$/year is lost. Exponential depreciation: $V_E(t)=24{,}000(0.90)^t$, where value drops 10% per year. Values are in dollars. Linear change subtracts the same amount yearly, while exponential change multiplies by the same factor yearly. In the context of this scenario, which model predicts a greater decrease after 3 years?
6. An investment grows continuously: $A(t)=2000e^{0.045t}$. When will it reach $5000$ dollars?
7. A harbor tide height (meters) follows $H(t)=2.6\sin\!\left(\frac{\pi}{6}t\right)+3.1$, where $t$ is hours after midnight. The midline is $3.1$ meters, and the tide oscillates symmetrically about this level. The amplitude represents half the high-to-low tidal range. Using the given function, calculate the amplitude of the function given in the scenario.
8. A car's value is $\$18{,}500$ and depreciates 7% yearly, modeled by $V(t)=18500(0.93)^t$. Using the information, what will the value be after 3 years?
9. Using the matrix provided, how does matrix multiplication affect $\mathbf{v}$ if $\mathbf{A}=\begin{pmatrix}1&2\\0&1\end{pmatrix}$ and $\mathbf{v}=\begin{pmatrix}1\\3\end{pmatrix}$?​
10. For what values of $x$ in the interval $[0, 2\pi]$ is the inequality $\sqrt{2}\cos(x) + 1 \le 0$ satisfied?
11. A ship travels from point $A$ to $B$ (12 km), then from $B$ to $C$ (5 km). The direct distance from $A$ to $C$ is 13 km, forming triangle $ABC$. The deviation angle at $B$ is $u$. Use $u=\arccos\!\left(\frac{AB^2+BC^2-AC^2}{2\cdot AB\cdot BC}\right)$ to find $u$ in degrees.
12. Based on the logarithmic expression provided, simplify $\log_2(8m)-\log_2\!\left(\dfrac{m}{4}\right)$ for $m>0$.
13. What is the physical meaning of the value -10 in the model?
14. A football is kicked from ground with $V_0=30\text{ m/s}$: $x(t)=30\cos\theta\,t$, $y(t)=30\sin\theta\,t-4.9t^2$. How does increasing $\theta$ affect trajectory?
15. The function $k$ is given by $k(t) = 50(2)^{-t}$. Which of the following statements is true?
16. A graphics filter shears horizontally using $H=[1\ 2;\ 0\ 1]$. Using the given transformation, what is $H\langle 1,3\rangle$?
17. For the polar equation $r=1+\sin(\theta)$, what special curve is represented?​
18. A projectile is $\vec{r}(t)=\langle 18t,\,2+12t-4.9t^2\rangle$; determine $\vec{r}(1)$.
19. A town's population rises after a new highway opens, but land and water limits slow later growth. Model 1: Exponential, $y=12000(1.05)^t$ (growth factor $1.05$). Model 2: Logarithmic, $y=10000+3500\log(0.5t+1)$ (coefficient $3500$). Early years show near-constant percent increases, but later years add fewer people each year. Using the information from the passage, which model best predicts long-term behavior in the context described?
20. A cup of coffee is $T_0=90^\circ\text{C}$ at $t=0$ minutes in a $20^\circ\text{C}$ room. A linear model cools by $1.5^\circ\text{C}$/min: $T_L(t)=90-1.5t$. An exponential model follows Newton-style cooling: $T_E(t)=20+70(0.92)^t$. Variables: $t$ in minutes, $T$ in $^\circ\text{C}$. Linear cooling drops by the same number of degrees each minute, while exponential cooling slows as it approaches room temperature. Using the information provided, what assumptions are made in the linear model vs the exponential model?

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