Linear Model Word Problems - Accuplacer Qas

A small business finds that its profit $P$ in dollars is given by $P(x) = 2.50x - 400$ , where $x$ is the number of items sold. If the business wants to make a profit of exactly $1000, how many items must it sell?
Answer: 560 The profit model is $P(x) = 2.50x - 400$ . We are given that the desired profit is $1000, so we set $P(x) = 1000$ . This gives the equation $1000 = 2.50x - 400$ . To solve for $x$ , first add 400 to both sides: $1400 = 2.50x$ . Then, divide both sides by 2.50: $x = 1400 / 2.50 = 560$ . The business must sell 560 items.
The number of trees, $N$ , on a tree farm is modeled by the equation $N(t) = 4000 - 150t$ , where $t$ is the number of years since 2020. What is the correct interpretation of the number -150 in this model?
Answer: The number of trees on the farm decreases by 150 each year. In the linear model $N(t) = 4000 - 150t$ , which is in the form $y = b + mx$ , the coefficient of the variable $t$ is the slope, which represents the rate of change. In this context, $t$ is the number of years, and $N(t)$ is the number of trees. The slope is -150, which means that the number of trees is changing by -150 each year. In other words, the number of trees decreases by 150 each year.
A salesperson earns a base salary of $1,200 per month plus a 5% commission on all sales. If the salesperson earned a total of $3,500 in one month, what were their total sales for that month?
Answer: $46,000 Let $S$ be the total sales. The total earnings are modeled by the equation: Earnings = Base Salary + Commission. The commission is 5% of sales, or $0.05S$ . So, $3500 = 1200 + 0.05S$ . First, subtract the base salary from the total earnings to find the commission amount: $3500 - 1200 = 2300$ . So, ($2,300) is the commission earned. Now, solve for sales: $2300 = 0.05S$ . Divide both sides by 0.05: $S = 2300 / 0.05 = 46000$ . The total sales were $46,000.
A delivery service uses the model $T = 15 + 3.5d$ to calculate delivery time $T$ (in minutes) based on distance $d$ (in miles). Due to traffic conditions, actual delivery times are consistently 20% longer than the model predicts. If a delivery actually took 54 minutes, what was the distance traveled?
Answer: 9.0 miles This problem tests your ability to work backwards through a multi-step relationship involving percentages and linear equations. When you see "actual times are 20% longer than predicted," you need to first find the predicted time, then use the model to find the distance. Since actual delivery times are 20% longer than predicted, the actual time of 54 minutes represents 120% of the predicted time. To find the predicted time: $\text{Predicted time} = \frac{54}{1.20} = 45 \text{ minutes}$ Now use the model $T = 15 + 3.5d$ with $T = 45$ : $45 = 15 + 3.5d$ $30 = 3.5d$ $d = \frac{30}{3.5} = \frac{60}{7} ≈ 8.57 \text{ miles}$ Rounding to the nearest half-mile gives us 9.0 miles, which is answer D. Let's examine why the other answers are wrong. Choice A (7.5 miles) comes from incorrectly subtracting 20% from 54 instead of dividing by 1.20. Choice B (11.1 miles) results from using 54 minutes directly in the formula without accounting for the 20% increase. Choice C (10.5 miles) appears to stem from miscalculating the percentage adjustment or making an arithmetic error in the division. Strategy tip: When dealing with percentage increases in real-world scenarios, always identify what the given value represents (actual vs. predicted) and work backwards methodically. Don't rush to plug numbers into formulas—first ensure you're using the right values.
A water tank is being drained at a constant rate. After 12 minutes, the tank contains 450 gallons. After 20 minutes, it contains 330 gallons. If the tank continues to drain at this rate, after how many total minutes will the tank contain exactly 75 gallons?
Answer: 37 minutes The rate of drainage is $(450-330)÷(20-12) = 120÷8 = 15$ gallons per minute. Using point-slope form with $(12, 450)$ : $V = 450 - 15(t-12) = 450 - 15t + 180 = 630 - 15t$ . Setting $V = 75$ : $75 = 630 - 15t$ , so $15t = 555$ and $t = 37$ minutes. Choice A uses an incorrect initial condition. Choice B results from calculation errors in the rate. Choice D results from using the wrong reference point.
A plant is currently 12 cm tall and grows at a rate of 0.25 cm per day. In how many days will the plant's height be double its current height?
Answer: 48 The current height of the plant is 12 cm. Double its current height is $12 \times 2 = 24$ cm. Let $d$ be the number of days. The height of the plant after $d$ days can be modeled by the equation $H(d) = 12 + 0.25d$ . We want to find the value of $d$ when $H(d) = 24$ . So, we solve the equation $12 + 0.25d = 24$ . Subtract 12 from both sides: $0.25d = 12$ . Divide both sides by 0.25 (or multiply by 4): $d = 12 / 0.25 = 48$ . It will take 48 days.
A printing company charges a one-time setup fee plus a fixed price per T-shirt. The cost for 25 T-shirts is $200, and the cost for 75 T-shirts is $500. What would be the total cost for an order of 110 T-shirts?
Answer: $710 First, find the price per T-shirt (the slope). The change in cost is $\$500 - \$200 = \$300$ . The change in quantity is $75 - 25 = 50$ T-shirts. So the price per shirt is $\$300 / 50 = \$6$ . Now, find the setup fee (the y-intercept). Using the first data point: $25 \text{ shirts} \times \$6/\text{shirt} = \$150$ . The total cost was $200, so the setup fee is $\$200 - \$150 = \$50$ . The linear model is $C = 6n + 50$ , where $n$ is the number of shirts. For 110 shirts, the cost is $C = 6(110) + 50 = 660 + 50 = \$710$ .
The monthly cost for a streaming service is $12. Members can also rent premium movies for $4 each. A member's total spending in a month was between $25 and $45, exclusive of the endpoints. Which of the following is a possible number of premium movies the member rented?
Answer: 8 Let $n$ be the number of premium movies rented. The total cost is $12 + 4n$ . The problem states that the spending is between $25 and $45, so we have the compound inequality $25 < 12 + 4n < 45$ . First, subtract 12 from all parts: $25 - 12 < 4n < 45 - 12$ , which simplifies to $13 < 4n < 33$ . Next, divide all parts by 4: $13/4 < n < 33/4$ , which is $3.25 < n < 8.25$ . Since $n$ must be a whole number, the possible values for $n$ are 4, 5, 6, 7, and 8. Of the given choices, only 8 falls within this range.
A car rental company charges a flat fee of $40 plus $0.25 per mile. A second company charges a flat fee of $25 plus $0.40 per mile. At how many miles will the cost to rent a car from both companies be the same?
Answer: 100 Let $m$ be the number of miles. The cost for the first company is $C_1 = 40 + 0.25m$ . The cost for the second company is $C_2 = 25 + 0.40m$ . To find when the costs are the same, set $C_1 = C_2$ : $40 + 0.25m = 25 + 0.40m$ . Subtract $25$ from both sides: $15 + 0.25m = 0.40m$ . Subtract (0.25m) from both sides: $15 = 0.15m$ . Divide by $0.15$ : $m = 15 / 0.15 = 100$ . The cost will be the same at 100 miles.
A car's fuel tank holds 18 gallons of gasoline, and the car has a fuel efficiency of 25 miles per gallon. Which equation represents the amount of gasoline $g$ , in gallons, remaining in the tank after driving $d$ miles, assuming the tank was full at the start?
Answer: $g = 18 - \frac{d}{25}$ The initial amount of gasoline is 18 gallons. To find the amount of gasoline used, divide the number of miles driven, $d$ , by the fuel efficiency, 25 miles per gallon. So, gas used = $\frac{d}{25}$ . The amount of gasoline remaining, $g$ , is the initial amount minus the amount used. Therefore, the correct equation is $g = 18 - \frac{d}{25}$ .
A piece of machinery was purchased for $50,000. Its value depreciates linearly, and it has a value of $35,000 after 3 years. What will its value be after 7 years?
Answer: $15,000 First, calculate the annual depreciation rate. The value decreased by $\$50,000 - \$35,000 = \$15,000$ over 3 years. The annual depreciation is $\$15,000 / 3 \text{ years} = \$5,000$ per year. The value $V$ after $t$ years is $V(t) = 50000 - 5000t$ . To find the value after 7 years, calculate $V(7) = 50000 - 5000(7) = 50000 - 35000 = \$15,000$ .
A company's weekly profit, $P$ , is modeled by the equation $P(n) = 30n - 1,500$ , where $n$ is the number of units sold. What is the minimum number of whole units the company must sell to make a profit?
Answer: 51 To make a profit, the profit $P$ must be greater than zero. So we set up the inequality $30n - 1500 > 0$ . Add 1,500 to both sides: $30n > 1500$ . Divide both sides by 30: $n > 50$ . Since $n$ must be a whole number of units, the smallest integer greater than 50 is 51. The company must sell at least 51 units to make a profit.
A car rental company charges a flat fee of $35 plus $0.28 per mile driven. If Maria's total rental cost was $91.40, and she received a $10\%$ discount on the mileage charges only, how many miles did she drive?
Answer: 200 miles Let $m$ = miles driven. The mileage charge before discount is $0.28m$ . With a 10% discount on mileage, she pays $0.9(0.28m) = 0.252m$ for mileage. Total cost: $35 + 0.252m = 91.40$ . Solving: $0.252m = 56.40$ , so $m = 56.40 ÷ 0.252 = 200$ miles. Choice A results from using the discounted rate incorrectly. Choice C results from applying the discount to the total cost. Choice D results from not applying the discount at all.
A gym membership costs $60 for an initiation fee and $35 per month. If a member pays for 2 years of membership upfront, what is their average cost per month over the 2-year period?
Answer: $37.50 First, calculate the total number of months in 2 years: $2 \times 12 = 24$ months. Next, calculate the total cost of the monthly fees over this period: $24 \text{ months} \times \$35/\text{month} = \$840$ . Then, add the one-time initiation fee to find the total cost: $\$840 + \$60 = \$900$ . Finally, to find the average cost per month, divide the total cost by the number of months: $\$900 / 24 \text{ months} = \$37.50$ per month.
A mobile phone plan costs $30 per month, which includes 4 gigabytes (GB) of data. There is an additional charge of $8 for each gigabyte of data used over the included 4 GB. If a customer's bill for one month was $62, how many total gigabytes of data did they use?
Answer: 8 First, find the amount of the bill that was for extra data by subtracting the base monthly cost: $\$62 - \$30 = \$32$ . Next, determine how many extra gigabytes of data this amount paid for. Since each extra gigabyte costs $8, divide the extra charge by the cost per gigabyte: $\$32 / \$8/GB = 4$ GB. This is the amount of data used in addition to the included amount. To find the total data used, add this extra amount to the included amount: $4 \text{ GB} + 4 \text{ GB} = 8$ GB.
A smartphone's battery is at 92% and drains at a rate of 12 percentage points per hour of constant use. If the user needs the phone to have at least 20% battery remaining, for how many hours at most can they use the phone?
Answer: 6.00 Let $h$ be the number of hours of use. The battery percentage $P$ can be modeled as $P(h) = 92 - 12h$ . The user wants the battery to be at least 20%, so we set up the inequality $92 - 12h \ge 20$ . Subtract 92 from both sides: $-12h \ge -72$ . Divide by -12 and reverse the inequality sign: $h \le (-72) / (-12)$ , which simplifies to $h \le 6$ . The maximum number of hours is 6.
An online tutoring service charges an initial setup fee plus an hourly rate. After 5 hours of tutoring, the total cost is $85. After 9 hours, the total cost is $125. What would be the cost for exactly 15 hours of tutoring?
Answer: $205 for 15 hours of tutoring sessions The hourly rate is $(125-85)÷(9-5) = 40÷4 = \$10$ per hour. Using the point (5, 85): $C = 85 + 10(h-5) = 85 + 10h - 50 = 35 + 10h$ . For 15 hours: $C = 35 + 10(15) = 35 + 150 = \$205$ . Choice A results from using an incorrect setup fee. Choice B results from calculation errors in the slope. Choice D results from errors in applying the linear model.
A hiker is descending a trail. At 1:00 PM, their elevation is 7,200 feet. At 3:30 PM, their elevation is 5,700 feet. If the hiker descends at a constant rate, what was their elevation at noon (12:00 PM)?
Answer: 7,800 feet First, calculate the rate of descent. The time elapsed between 1:00 PM and 3:30 PM is 2.5 hours. The change in elevation is $7200 - 5700 = 1500$ feet. The rate is $1500 \text{ feet} / 2.5 \text{ hours} = 600$ feet per hour. Noon is one hour before 1:00 PM. Since the hiker is descending, their elevation at noon would have been higher than at 1:00 PM. To find the elevation at noon, add one hour's worth of descent to the 1:00 PM elevation: $7200 \text{ feet} + 600 \text{ feet} = 7800$ feet.
A printing company charges based on the linear model $C = 45 + 0.12n$ , where $C$ is the total cost in dollars and $n$ is the number of pages printed. If a customer's budget allows for a maximum cost of $93, but they need at least 250 pages printed, what is the maximum number of additional pages they can afford beyond the minimum required?
Answer: 150 additional pages beyond the required 250 pages With budget $93: $93 = 45 + 0.12n$ , so $0.12n = 48$ and $n = 400$ total pages possible. Since 250 pages are required minimum, the additional pages possible are $400 - 250 = 150$ . Choice B gives the total pages but doesn't answer the question about additional pages. Choice C and D result from calculation errors in solving the linear equation.
A taxi fare includes a $2.75 flag-drop fee plus $2.20 per mile. A passenger has at most $30 to spend on the ride. What is the maximum whole number of miles the passenger can travel?
Answer: 12 Let $m$ be the number of miles. The total cost $C$ is given by $C = 2.75 + 2.20m$ . The passenger has at most $30, so $2.75 + 2.20m \le 30$ . Subtract 2.75 from both sides: $2.20m \le 27.25$ . Divide both sides by 2.20: $m \le 27.25 / 2.20 \approx 12.386$ . Since the question asks for the maximum whole number of miles, the passenger can travel 12 miles without exceeding the $30 limit.

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