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Linear Models

Grade 8 · Algebra · Worksheet 3

  1. Aroha runs a small home-based business making and selling handmade candles. She finds that her total cost C (in dollars) to produce x candles can be modeled by the linear equation C = 3x + 15. In this equation, what does the number 3 represent in the context of Aroha's candle business, and what does the number 15 represent? Answer: ______________
  2. 3² × (5 - 2) + √16 = ? Answer: ______________
  3. Isabella is tracking the growth of her sunflower plant. She finds that the height of the plant can be modeled by the equation h = 2.7w + 12, where h is the height in centimeters and w is the number of weeks since she planted the seed. What does the number 2.7 represent in this context, and what does the number 12 represent? Answer: ______________
  4. Mere runs a small catering business. The total cost C in dollars to cater for x people at an event is given by the linear equation C = 12x + 80. Interpret the meaning of the slope (12) and the y-intercept (80) in the context of this catering business. Answer: ______________
  5. A rectangular prism is drawn with dimensions 12 cm by 8 cm by 5 cm. A smaller similar prism is created by applying a scale factor of 1/2 to all dimensions. What is the ratio of the surface area of the smaller prism to the surface area of the original prism?
    Answer: ______________
  6. (3x + 12) ÷ 4 = 9 Answer: ______________
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Answer Key & Explanations

Linear Models · Grade 8 · Worksheet 3

  1. Aroha runs a small home-based business making and selling handmade candles. She finds that her total cost C (in dollars) to produce x candles can be modeled by the linear equation C = 3x + 15. In this equation, what does the number 3 represent in the context of Aroha's candle business, and what does the number 15 represent? Answer: The 3 represents the cost in dollars to make each candle (the variable cost per candle). The 15 represents the fixed startup or overhead cost in dollars that Aroha must pay regardless of how many candles she makes (such as equipment, packaging, or a workspace fee). Solution: The equation is C = 3x + 15, where C is total cost in dollars and x is the number of candles produced. The number 3 is the coefficient of x.
    Full step-by-step solution

    Step 1: The equation is C = 3x + 15, where C is total cost in dollars and x is the number of candles produced. Step 2: The number 3 is the coefficient of x. This means that for every additional candle Aroha makes, the total cost increases by $3. So, 3 represents the cost per candle (the variable cost). Step 3: The number 15 is the constant term. When x = 0 (no candles are made), C = 3(0) + 15 = 15. This means that even if Aroha produces zero candles, she still has a cost of $15. This is the fixed cost (e.g., supplies, equipment rental, or packaging). Step 4: Therefore, in context: the 3 means each candle costs $3 to make, and the 15 means Aroha has a fixed cost of $15 regardless of production.

  2. 3² × (5 - 2) + √16 = ? Answer: 31 Solution: Calculate inside the parentheses: 5 - 2 = 3 Calculate the exponent: 3² = 9 Multiply: 9 × 3 = 27 Calculate the square root: √16 = 4 Add: 27 + 4 = 31 The answer is 31.
    Full step-by-step solution

    Step 1: Calculate inside the parentheses: 5 - 2 = 3 Step 2: Calculate the exponent: 3² = 9 Step 3: Multiply: 9 × 3 = 27 Step 4: Calculate the square root: √16 = 4 Step 5: Add: 27 + 4 = 31 The answer is 31.

  3. Isabella is tracking the growth of her sunflower plant. She finds that the height of the plant can be modeled by the equation h = 2.7w + 12, where h is the height in centimeters and w is the number of weeks since she planted the seed. What does the number 2.7 represent in this context, and what does the number 12 represent? Answer: The 2.7 represents the growth rate of the sunflower in centimeters per week, and the 12 represents the initial height of the sunflower in centimeters when it was planted. Solution: Identify the form of the equation. The equation h = 2.7w + 12 is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. Interpret the slope (m = 2.7).
    Full step-by-step solution

    Step 1: Identify the form of the equation. The equation h = 2.7w + 12 is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. Step 2: Interpret the slope (m = 2.7). The slope represents the rate of change. Since w is weeks and h is height, the slope 2.7 means the plant grows 2.7 centimeters per week. Step 3: Interpret the y-intercept (b = 12). The y-intercept is the value of h when w = 0. When w = 0 (at the time of planting), the height is 12 centimeters. This is the initial height of the plant. Step 4: Final answer: The 2.7 represents the growth rate of the sunflower in centimeters per week, and the 12 represents the initial height of the sunflower in centimeters when it was planted.

  4. Mere runs a small catering business. The total cost C in dollars to cater for x people at an event is given by the linear equation C = 12x + 80. Interpret the meaning of the slope (12) and the y-intercept (80) in the context of this catering business. Answer: The slope of 12 means the cost increases by $12 for each additional person catered for. The y-intercept of 80 means the fixed cost (such as kitchen setup, equipment rental, or delivery fee) is $80, even if no one attends. Solution: The equation is C = 12x + 80, where C is total cost in dollars and x is number of people. The slope is 12. In this context, it represents the cost per additional person.
    Full step-by-step solution

    Step 1: The equation is C = 12x + 80, where C is total cost in dollars and x is number of people. Step 2: The slope is 12. In this context, it represents the cost per additional person. For every 1 more person catered for, the total cost increases by $12 (this could cover food, plates, napkins, etc. per person). Step 3: The y-intercept is 80. This is the value of C when x = 0. It represents the fixed costs that are charged even if no people attend (e.g., kitchen rental, delivery fee, or basic service charge). Answer: The slope 12 means each additional person adds $12 to the total cost. The y-intercept 80 means the fixed cost is $80 regardless of the number of people.

  5. A rectangular prism is drawn with dimensions 12 cm by 8 cm by 5 cm. A smaller similar prism is created by applying a scale factor of 1/2 to all dimensions. What is the ratio of the surface area of the smaller prism to the surface area of the original prism? Answer: 1/4 Solution: Original prism dimensions: 12 cm × 8 cm × 5 cm Scale factor for dimensions: 1/2 Surface area scales by the square of the scale factor Scale factor squared = (1/2)^2 = 1/4 Therefore, the ratio of surface areas is 1/4 The answer is 1/4.
    Full step-by-step solution

    Step 1: Original prism dimensions: 12 cm × 8 cm × 5 cm Step 2: Scale factor for dimensions: 1/2 Step 3: Surface area scales by the square of the scale factor Step 4: Scale factor squared = (1/2)^2 = 1/4 Step 5: Therefore, the ratio of surface areas is 1/4 The answer is 1/4.

  6. (3x + 12) ÷ 4 = 9 Answer: 8 Solution: The equation is (3x + 12) ÷ 4 = 9 Multiply both sides by 4 to eliminate the division: 3x + 12 = 9 × 4 Calculate 9 × 4 = 36, so 3x + 12 = 36 Subtract 12 from both sides: 3x = 36 - 12 Calculate 36 - 12 = 24, so 3x = 24 Divide both sides by 3: x = 24 ÷ 3 Calculate 24 ÷ 3 = 8, so x = 8 The answer is 8.
    Full step-by-step solution

    Step 1: The equation is (3x + 12) ÷ 4 = 9 Step 2: Multiply both sides by 4 to eliminate the division: 3x + 12 = 9 × 4 Step 3: Calculate 9 × 4 = 36, so 3x + 12 = 36 Step 4: Subtract 12 from both sides: 3x = 36 - 12 Step 5: Calculate 36 - 12 = 24, so 3x = 24 Step 6: Divide both sides by 3: x = 24 ÷ 3 Step 7: Calculate 24 ÷ 3 = 8, so x = 8 The answer is 8.