Practice Test 18 - Algebra 1

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

A rectangular garden has a length that is 4 feet more than twice its width. If the perimeter must be at least 32 feet but no more than 50 feet, which compound inequality represents the possible widths $w$ of the garden?
The function $f(t) = 250(1.08)^{3t}$ models a quantity over time $t$ measured in years. What is the effective annual growth rate, and what does the coefficient 3 in the exponent indicate?
Show the minimum by completing the square: $f(x) = x^2 - 10x + 30$ . What is the minimum value of $f(x)$ ?
A radioactive substance decays according to $N(t) = 800(0.5)^{t/12}$ , where $N$ is the number of grams remaining after $t$ hours. A researcher studying this substance wants to know when only 100 grams remain. In this decay model, what specific information does the fraction $\frac{t}{12}$ in the exponent provide about the substance's half-life?
In the rectangle area formula $A = lw$ (where $A$ is area, $l$ is length, and $w$ is width), solve for $w$ . Treat the other variables like numbers and use inverse operations as you would in a numeric equation.
A town's population is recorded each year: Year $t$ : 0, 1, 2, 3 Population $P(t)$ : 20,000; 21,600; 23,328; 25,194.24 From the data, determine if there is constant percent change. If so, what is the percent rate per year?
Use the geometric series sum formula $S_n = \frac{a(1-r^n)}{1-r}$ (for $r \ne 1$ ) to find the sum of the first $6$ terms of the geometric series $5 + 10 + 20 + 40 + 80 + 160.$ Clearly, $a=5$ , $r=2$ , and $n=6$ .
Two investment funds are compared over 5 years. Fund A grows at 6% annually, while Fund B grows at 3% every 6 months. Which statement correctly compares their growth?
Solve using elimination: $\begin{cases} 2x+3y=12\\ 2x-y=4 \end{cases}$ What is the solution $(x, y)$ ?
To keep exponent rules consistent, we want $b^{m/n}$ to satisfy both $b^{m/n}=(b^{1/n})^m$ and $b^{m/n}=(b^m)^{1/n}$ for $b>0$ . Which option correctly shows these give the same radical form?
A gym membership fee is tracked over months: Month $m$ : 0, 1, 2, 3 Fee $F(m)$ : $40, $45, $50, $55 Does this represent constant percent change (exponential) or constant additive change (linear)?
Complete the square to solve $x^2-4x+5=0$ for complex solutions. Use $i^2=-1$ and write the answers in $a\pm bi$ form.
Given $f(x)=2x-5$ and $g(x)=x^2+1$ , find $f(g(x))$ .
Complete the square to find the vertex of $g(x)=-2x^2+8x-1$ .
Which statement about growth rates is true for sufficiently large $x$ ?
A rideshare company charges a flat booking fee plus a per-mile charge. The total cost $C$ (in dollars) for a ride of $m$ miles is $C = 2.25m + 4.50$ . What does the 2.25 represent in this context?
In the surface area formula for a cylinder, one part can be written as $2\pi r(r+h).$ Which describes the structure of $2\pi r(r+h)$ ? (View $(r+h)$ as a single unit.)
Which recursive formula represents $T_n$ , the number of trees in the $n$ th row?
Rewrite $x^{2/3}$ using radicals.
How many solutions does the equation $y=-2x+5$ have? (Think about what its graph represents.)

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