Practice Test 13 - Algebra 2

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

Pipe A can drain a pool in 5 hours. Pipe B can fill the same pool in 7 hours. If both pipes are open at the same time, the pool is still draining overall. Set up and solve a rational equation to find how long it takes to drain one full pool. Let $t$ = time in hours to drain the pool with both pipes open.
The expression $3x^2y + xy^2 - 6x^2y + 2xy^2$ can be simplified by combining like terms. After simplification, what is the product of all the coefficients?
Points $z_1=a+bi$ and $z_2=c+di$ on the complex plane are separated by distance $|z_1-z_2|=\sqrt{(a-c)^2+(b-d)^2}$ , the same as the 2D distance formula. What is the distance between $z_1=5+2i$ and $z_2=1-3i$ ?
For $f(x)=\frac{x^2-9}{x^2-4x-5},$ identify the zeros and vertical asymptotes. (Factor to find features.)
Solve the exponential equation $3\cdot 10^{2x}=75$ for $x$ . Give the exact solution (as a logarithm) and an approximate value using technology.
An investment account has an annual interest rate of 8% compounded continuously. To find the equivalent quarterly compounding rate, which transformation of $e^{0.08t}$ is most useful?
To sketch $f(x)=x^2+2x-8$ , first factor to find the zeros. Which option gives the correct factored form and zeros (x-intercepts)?
Let $z_1=2+3i$ and $z_2=8+1i$ represent points on the complex plane. The midpoint is $\dfrac{z_1+z_2}{2}=\left(\dfrac{a+c}{2}\right)+\left(\dfrac{b+d}{2}\right)i$ , matching the coordinate midpoint formula. What is the midpoint?
Which set of points on the complex plane represents all complex numbers $z$ such that $|z|=5$ ? (Recall $|z|$ is the distance from the origin.)
Consider the system where $p(x) = x^3 - 2x^2 + 1$ and $q(x) = 2x - 1$ . A student claims that since both functions pass through the point $(1, 1)$ , the equation $x^3 - 2x^2 + 1 = 2x - 1$ has $x = 1$ as its only solution. What is wrong with this reasoning?
Use the change-of-base formula (optional) and log properties to rewrite $\log_4(25)$ in terms of natural logarithms. You may use $\log_b(x)=\frac{\ln(x)}{\ln(b)}.$
A bacteria culture starts with 500 cells and grows by 12% each hour. Let $P$ be the number of cells after $t$ hours. What equation models the situation, and what are appropriate axes labels to graph $P$ versus $t$ for $0\le t\le 10$ ?
A bacteria culture follows $N(t)=200\cdot 2^{t/6}$ . When the culture reaches $N=1500$ , what is $t$ ? Express your answer as a logarithm and approximate using a calculator.
Based on the table shown, what is the value of $g^{-1}(5)$ ?
A population is measured every year: Year 0: 40,000 Year 1: 41,600 Year 2: 43,264 Year 3: 44,994.56 Classify the pattern as exponential growth, exponential decay, or neither, and identify the percent rate per year.
Verify by composition whether $f(x)=\dfrac{1}{x-1}$ and $g(x)=1+\dfrac{1}{x}$ are inverses. Compute $f(g(x))$ and $g(f(x))$ and simplify (assume inputs avoid values that make denominators zero).
A bacteria culture starts with 80 bacteria and triples every hour. Let $a_n$ be the number of bacteria after $n-1$ hours (so $a_1=80$ ), with domain $n\in\{1,2,3,\dots\}$ . Model this situation with both a recursive definition and an explicit formula.
The rational function $r(x) = \frac{x^2 - 4}{x - 1}$ and the linear function $s(x) = x + 3$ appear to intersect at two points when graphed. However, when solving $\frac{x^2 - 4}{x - 1} = x + 3$ algebraically, why might the number of actual intersection points differ from what the graph suggests?
A clinic is assembling care packages with bandage packs (x) and antiseptic wipes (y). Each bandage pack costs $4 and each wipe pack costs $3. The clinic can spend at most $120: $4x+3y\le 120$ . Each package must include at least 10 total packs: $x+y\ge 10$ . Due to storage limits, there can be at most 25 wipe packs: $y\le 25$ . Also $x,y\ge 0$ . Is the plan $(x,y)=(15,25)$ viable?
The decibel level of a sound is given by $\text{dB}=10\log_{10}\left(\frac{I}{I_0}\right)$ , where $I$ is the sound intensity and $I_0$ is a reference intensity. If a sound has intensity $2.5\times 10^4\,I_0$ , find its decibel level. Round to the nearest tenth and interpret your result in context.

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