Modeling with Expressions - Pre Algebra

1. A car rental company charges a base fee plus an additional amount per mile driven. After driving 50 miles, the total cost is $45. After driving 120 miles, the total cost is $66. Which expression represents the total cost $C$ for driving $m$ miles?
   Answer: $C = 0.3m + 30$ To find the expression, we need to determine the base fee and rate per mile. Using the two data points: (50, 45) and (120, 66). The rate per mile is (66-45)/(120-50) = 21/70 = 0.3 dollars per mile. The base fee is 45 - 0.3(50) = 45 - 15 = 30 dollars. Therefore C = 0.3m + 30. Choice B uses the wrong base fee (45 instead of 30). Choice C switches the rate and base fee values. Choice D incorrectly places the base fee as the coefficient of m.
2. A manufacturing company's daily profit $P$ (in dollars) depends on the number of items $x$ produced according to the expression $P = -2x^2 + 80x - 300$. What do the units of each term represent in this context?
   Answer: $-2x^2$: decreasing profit effect in dollars; $80x$: revenue per item in dollars; $-300$: fixed daily costs in dollars In the profit equation P = -2x² + 80x - 300, each term must have units of dollars since P is in dollars. The -2x² term represents a decreasing profit effect (diminishing returns) in dollars, 80x represents revenue contribution per item totaling dollars, and -300 represents fixed daily costs in dollars. Choice A incorrectly states units for the quadratic term. Choice B doesn't recognize the fixed cost nature of the constant term. Choice D has incorrect units for the quadratic term and misidentifies the constant term as variable costs.
3. A pizza restaurant charges a delivery fee plus a cost per topping. A pizza with 2 toppings costs $14, and a pizza with 5 toppings costs $20. Which expression represents the cost $C$ of a pizza with $t$ toppings, and what constraint makes sense for the number of toppings?
   Answer: $C = 10 + 2t$ with $t \geq 0$ and $t$ is an integer When you encounter a word problem about costs with a fixed fee plus a variable rate, you're dealing with linear relationships. The key is identifying the fixed cost (y-intercept) and the rate of change (slope). Let's use the given information to find the cost structure. With 2 toppings costing $14 and 5 toppings costing $20, we can find the cost per topping by calculating the rate of change: $\frac{20 - 14}{5 - 2} = \frac{6}{3} = 2$ dollars per topping. Now we can find the delivery fee (fixed cost). Using the point (2, 14) and substituting into $C = \text{delivery fee} + 2t$: $14 = \text{delivery fee} + 2(2)$, so the delivery fee is $10. This gives us $C = 10 + 2t$. For constraints, since you can order a pizza with zero toppings (just cheese), $t \geq 0$ makes sense. Also, you can't have half a topping, so $t$ must be an integer. Answer A gives $C = 2t + 10$ (mathematically equivalent to our answer) but incorrectly allows fractional toppings since it doesn't specify $t$ is an integer. Answer B uses the wrong relationship $C = 6t + 2$, which would mean $2 per topping with a $2 delivery fee – this doesn't match our given data. Answer D has the same constraint error as A, not specifying that toppings must be whole numbers. Remember: In linear cost problems, always find the rate first using two points, then work backward to find the fixed cost. Also consider real-world constraints – can the variable realistically be fractional?
4. A rectangular garden has a length that is 8 feet more than twice its width. If the perimeter must be at least 76 feet to accommodate a fence, which inequality represents the constraint on the width $w$ in feet?
   Answer: $6w + 16 \geq 76$ The length is $2w + 8$ and the width is $w$. The perimeter formula is $P = 2(\text{length} + \text{width}) = 2(2w + 8 + w) = 2(3w + 8) = 6w + 16$. Since the perimeter must be at least 76 feet, we have $6w + 16 \geq 76$. Choice A only represents the length constraint, not perimeter. Choice C uses an incorrect coefficient for w. Choice D incorrectly calculates the perimeter expression.
5. A bakery sells cupcakes in boxes. Regular boxes hold 6 cupcakes and large boxes hold 12 cupcakes. If the bakery needs to package exactly 96 cupcakes using both types of boxes, and they must use at least 2 regular boxes, which equation represents this constraint where $r$ is regular boxes and $\ell$ is large boxes?
   Answer: $6r + 12\ell = 96$ with $r \geq 2$ and $r, \ell \geq 0$ When you encounter word problems involving multiple constraints, you need to translate each piece of information into mathematical language. This problem has three key pieces: the total number of cupcakes needed, the capacity of each box type, and the minimum requirement for regular boxes. Let's build the equation step by step. Since regular boxes hold 6 cupcakes and large boxes hold 12 cupcakes, and you need exactly 96 cupcakes total, the equation becomes $6r + 12\ell = 96$. The equal sign is crucial here because the problem states "exactly 96 cupcakes" - not "at most" or "at least." For the constraint "at least 2 regular boxes," this translates to $r \geq 2$. Since you can't have negative boxes, both $r$ and $\ell$ must be non-negative: $r, \ell \geq 0$. Looking at the wrong answers: Choice A uses $r > 2$, which means "more than 2 boxes," but the problem allows exactly 2 regular boxes. Choice B uses $6r + 12\ell \leq 96$, suggesting you need "at most" 96 cupcakes, but the problem requires exactly 96. Choice C uses $r + \ell = 96$, which ignores the different capacities of the boxes entirely - this would mean each box holds only 1 cupcake. Choice D correctly captures both the exact total requirement and the minimum constraint with $6r + 12\ell = 96$ and $r \geq 2$. Study tip: In constraint problems, distinguish between "at least" (≥), "at most" (≤), and "exactly" (=). These words directly determine which inequality or equation symbol to use.
6. A water tank is being filled at a constant rate. The tank starts with 150 gallons and gains 25 gallons every 4 minutes. Which expression gives the amount of water $W$ (in gallons) after $t$ minutes, and what does the coefficient of $t$ represent?
   Answer: $W = 150 + 6.25t$; coefficient represents gallons gained per minute The rate is 25 gallons per 4 minutes = 6.25 gallons per minute. So $W = 150 + 6.25t$ where the coefficient 6.25 represents gallons gained per minute. Choice A uses the wrong rate (25 instead of 6.25). Choice C has the correct expression but wrong interpretation of the coefficient. Choice D uses an incorrect rate calculation (25 × 4 = 100 instead of 25 ÷ 4 = 6.25).
7. A school is planning a field trip. The cost includes a $200 bus rental plus $15 per student for admission. If the school has a budget of at most $800 and needs at least 20 students to make the trip worthwhile, which system represents these constraints where $s$ is the number of students?
   Answer: $200 + 15s \leq 800$ and $s \geq 20$ and $s > 0$ When you encounter word problems involving constraints and inequalities, you need to translate each condition into mathematical language and determine whether boundaries are included or excluded. Let's break down the costs: there's a fixed $200 bus rental plus $15 per student. So the total cost is $200 + 15s$ where $s$ is the number of students. Since the budget is "at most $800," this means the cost can equal $800 or be less, giving us $200 + 15s \leq 800$. The phrase "at least 20 students" means 20 or more students are needed, so $s \geq 20$. Finally, since we're dealing with a real situation, we need a positive number of students, so $s > 0$. Choice A incorrectly omits the $200 bus rental from the cost equation, only including the per-student admission fee. Choice B mistakenly multiplies 200 by $s$, treating the bus rental as a per-student cost rather than a fixed expense. Choice C uses strict inequalities ($<$ and $>$) instead of inclusive ones, which would exclude the boundary values of exactly $800 budget and exactly 20 students – but the problem allows these exact amounts. Choice D correctly captures all constraints: $200 + 15s \leq 800$ (total cost within budget), $s \geq 20$ (minimum students required), and $s > 0$ (positive number of students). Study tip: In constraint problems, pay close attention to phrases like "at most," "at least," "more than," and "less than" – they determine whether you use $\leq, \geq, <,$ or $>$. Also, always identify fixed costs versus variable costs when setting up cost equations.
8. A movie theater offers two pricing plans. Plan A charges $12 per ticket. Plan B charges a $15 membership fee plus $8 per ticket. For what number of tickets $t$ is Plan B less expensive than Plan A?
   Answer: $t > 3.75$, so 4 or more tickets Plan A costs $12t$ and Plan B costs $15 + 8t$. For Plan B to be less expensive: $15 + 8t < 12t$, which gives $15 < 4t$, so $t > 3.75$. Since we need whole tickets, this means 4 or more tickets. Choice B uses incorrect arithmetic (dividing 15 by 8 instead of 4). Choice C reverses the inequality direction. Choice D incorrectly sets up the inequality as $8t > 15 + 12t$.

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