Interpret Expressions

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Answer Key & Explanations

Interpret Expressions · Grade 9 · Worksheet 3

1. The total cost C (in dollars) for Jordan's school club to produce 138 custom T-shirts is given by C = 7x + 125. In this expression, what quantity does the term 125 represent? Answer: 125 Solution: The cost function is C = 7x + 125, where x is the number of T-shirts. The term 7x represents the variable cost: $7 per shirt times x shirts. The constant term 125 does not depend on x.
   Full step-by-step solution

   Step 1: The cost function is C = 7x + 125, where x is the number of T-shirts. Step 2: The term 7x represents the variable cost: $7 per shirt times x shirts. Step 3: The constant term 125 does not depend on x. It represents the fixed costs (setup, design, equipment) that are incurred even if no shirts are produced. Step 4: Therefore, 125 represents the fixed cost of $125. Answer: $125 fixed cost.

2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (5,0), and (0,12). A circle is circumscribed around this triangle such that all three vertices lie on the circle's circumference. What is the equation of this circumscribed circle? Answer: (x-2.5)^2+(y-6)^2=42.25 Solution: Identify that this is a right triangle with legs along the axes, so the hypotenuse is from (5,0) to (0,12). For a right triangle, the hypotenuse is the diameter of the circumscribed circle.
   Full step-by-step solution

   Step 1: Identify that this is a right triangle with legs along the axes, so the hypotenuse is from (5,0) to (0,12). Step 2: For a right triangle, the hypotenuse is the diameter of the circumscribed circle. Step 3: Find the midpoint of the hypotenuse to get the center of the circle: ((5+0)/2, (0+12)/2) = (2.5, 6). Step 4: Calculate the length of the hypotenuse using the distance formula: sqrt((5-0)^2 + (0-12)^2) = sqrt(25 + 144) = sqrt(169) = 13. Step 5: The radius is half the diameter: 13/2 = 6.5. Step 6: Write the equation of the circle: (x-2.5)^2 + (y-6)^2 = (6.5)^2. Step 7: Calculate (6.5)^2 = 42.25. The equation is (x-2.5)^2 + (y-6)^2 = 42.25.

3. Isabella's company profit is modeled by P(t) = -2t² + 14t + 11, where t represents the number of years since the company started. What quantity does the constant term 11 represent in this context? Answer: 11 Solution: The profit function is P(t) = -2t² + 14t + 11, where t is the number of years since the company started. The constant term is 11, which does not depend on t.
   Full step-by-step solution

   Step 1: The profit function is P(t) = -2t² + 14t + 11, where t is the number of years since the company started. Step 2: The constant term is 11, which does not depend on t. Step 3: When t = 0 (the year the company started), P(0) = -2(0)² + 14(0) + 11 = 11. Step 4: This means the constant term represents the initial profit (or starting profit) when the company first began, before any changes due to time. Step 5: Therefore, the constant term 11 represents the profit (in appropriate monetary units) at the start of the company. The answer is 11.

4. A right triangle is inscribed in a circle such that its hypotenuse is the diameter of the circle. The legs of the triangle have lengths of 6 cm and 8 cm. What is the area of the circle? (Use π = 3.14) Answer: 78.5 Solution: A right triangle is inscribed in a circle with its hypotenuse as the diameter. The legs are 6 cm and 8 cm. We need the area of the circle.
   Full step-by-step solution

   Step 1: Understand the problem A right triangle is inscribed in a circle with its hypotenuse as the diameter. The legs are 6 cm and 8 cm. We need the area of the circle. Step 2: Find the hypotenuse For a right triangle with legs 6 cm and 8 cm, use the Pythagorean theorem: Hypotenuse^2 = 6^2 + 8^2 Hypotenuse^2 = 36 + 64 Hypotenuse^2 = 100 Hypotenuse = sqrt(100) = 10 cm. Step 3: Relate hypotenuse to the circle The hypotenuse is the diameter of the circle, so: Diameter = 10 cm. Step 4: Find the radius Radius = Diameter / 2 = 10 / 2 = 5 cm. Step 5: Area of the circle formula Area = π * r^2 Area = 3.14 * (5)^2 Area = 3.14 * 25 Area = 78.5 cm^2. Step 6: Conclusion The area of the circle is 78.5 cm^2.

5. Liam is designing a rectangular garden with a perimeter of 40 meters. He wants to model the area A of the garden as a function of its width w. Write the quadratic function that represents the area of the garden in terms of w, and determine the maximum possible area. Answer: A(w) = -w^2 + 20w; Maximum area = 100 m² Solution: We have a rectangular garden with perimeter \( P = 40 \) m. Let \( w \) = width (in meters) and \( l \) = length (in meters).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a rectangular garden with perimeter \( P = 40 \) m. Let \( w \) = width (in meters) and \( l \) = length (in meters). Perimeter formula for a rectangle: \[ P = 2l + 2w \] Substitute \( P = 40 \): \[ 2l + 2w = 40 \] --- **Step 2: Solve for length in terms of width** Divide both sides by 2: \[ l + w = 20 \] So: \[ l = 20 - w \] --- **Step 3: Write area as a function of \( w \)** Area \( A = l \times w \): \[ A(w) = (20 - w) \times w \] \[ A(w) = 20w - w^2 \] Rewriting in standard quadratic form: \[ A(w) = -w^2 + 20w \] --- **Step 4: Find the maximum possible area** This is a quadratic function with \( a = -1 \), \( b = 20 \), \( c = 0 \). Since \( a < 0 \), the parabola opens downward, so the vertex gives the maximum. Vertex \( w \)-coordinate: \[ w = -\frac{b}{2a} = -\frac{20}{2(-1)} = -\frac{20}{-2} = 10 \] --- **Step 5: Compute maximum area** Substitute \( w = 10 \) into \( A(w) \): \[ A(10) = - (10)^2 + 20(10) = -100 + 200 = 100 \] So maximum area is \( 100 \) m². --- **Final Answer:** \[ A(w) = -w^2 + 20w \] Maximum area = 100 m²

6. Noah's company profit is modeled by P(x) = 6x² - 11x + 16 where x represents hundreds of items sold. What does the constant term 16 represent in this context? Answer: 16 Solution: In the profit function P(x) = 6x² - 11x + 16, the variable x represents hundreds of items sold. The constant term 16 is independent of x, meaning it does not change based on the number of items sold.
   Full step-by-step solution

   Step 1: In the profit function P(x) = 6x² - 11x + 16, the variable x represents hundreds of items sold. Step 2: The constant term 16 is independent of x, meaning it does not change based on the number of items sold. Step 3: When x = 0 (no items sold), the profit is P(0) = 6(0)² - 11(0) + 16 = 16. Step 4: In a profit function, the constant term typically represents fixed costs or initial profit/loss that occurs regardless of sales. Step 5: Since the constant is positive, it represents the profit (in appropriate monetary units) when zero items are sold. The answer is 16.

7. Emma is analyzing the growth of a bacterial culture in her biology lab. The population P(t) after t hours is modeled by the exponential function P(t) = 500 * 2^(t/3). She needs to determine how long it will take for the population to reach 4000 bacteria. Which equation correctly represents this situation? Answer: B. 500 * 2^(t/3) = 4000 Solution: The given function is P(t) = 500 * 2^(t/3) We want to find when P(t) = 4000 Set up the equation: 500 * 2^(t/3) = 4000 Divide both sides by 500: 2^(t/3) = 8 Recognize that 8 = 2^3, so: 2^(t/3) = 2^3 Set exponents equal: t/3 = 3 Multiply both sides by 3: t = 9 The correct equation is 500 * 2^(t/3)…
   Full step-by-step solution

   Step 1: The given function is P(t) = 500 * 2^(t/3) Step 2: We want to find when P(t) = 4000 Step 3: Set up the equation: 500 * 2^(t/3) = 4000 Step 4: Divide both sides by 500: 2^(t/3) = 8 Step 5: Recognize that 8 = 2^3, so: 2^(t/3) = 2^3 Step 6: Set exponents equal: t/3 = 3 Step 7: Multiply both sides by 3: t = 9 Step 8: The correct equation is 500 * 2^(t/3) = 4000, which corresponds to choice C.

8. A scientist is studying bacterial growth in a lab culture. The population P(t) after t hours is modeled by the exponential function P(t) = 500 * 2^(t/3). If the scientist needs to determine when the population will reach 4000 bacteria, how many hours will it take? Answer: 9 Solution: Set up the equation using the given function: 500 * 2^(t/3) = 4000 Divide both sides by 500: 2^(t/3) = 8 Recognize that 8 = 2^3, so 2^(t/3) = 2^3 Since the bases are equal, set the exponents equal: t/3 = 3 Multiply both sides by 3: t = 9 Check: 500 * 2^(9/3) = 500 * 2^3 = 500 * 8 = 4000 The…
   Full step-by-step solution

   Step 1: Set up the equation using the given function: 500 * 2^(t/3) = 4000 Step 2: Divide both sides by 500: 2^(t/3) = 8 Step 3: Recognize that 8 = 2^3, so 2^(t/3) = 2^3 Step 4: Since the bases are equal, set the exponents equal: t/3 = 3 Step 5: Multiply both sides by 3: t = 9 Step 6: Check: 500 * 2^(9/3) = 500 * 2^3 = 500 * 8 = 4000 The answer is 9 hours.