Sketch Function Graphs

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Sketch Function Graphs · Grade 8 · Worksheet 2

1. A right triangle is drawn on a coordinate plane with vertices at (0,0), (12,0), and (12,5). A rectangle is inscribed inside this triangle such that one side lies along the x-axis from (0,0) to (x,0), and the opposite vertices touch the hypotenuse of the triangle. What is the area of the largest possible rectangle that can be inscribed in this triangle under these conditions? Answer: 30 Solution: Find the equation of the hypotenuse. The hypotenuse connects (0,0) and (12,5). The slope is (5-0)/(12-0) = 5/12.
   Full step-by-step solution

   Step 1: Find the equation of the hypotenuse. The hypotenuse connects (0,0) and (12,5). The slope is (5-0)/(12-0) = 5/12. The equation is y = (5/12)x. Step 2: Let the rectangle have width x along the x-axis. The top-right corner touches the hypotenuse at (x, y), where y = (5/12)x. Step 3: The rectangle's dimensions are width = x and height = y = (5/12)x. Step 4: The area function is A(x) = width × height = x × (5/12)x = (5/12)x². Step 5: The maximum occurs at the maximum x-value, which is when the rectangle spans the full base of the triangle. The triangle's base is from (0,0) to (12,0), so maximum x = 12. Step 6: Calculate the maximum area: A(12) = (5/12) × 12² = (5/12) × 144 = 5 × 12 = 30. The answer is 30.

2. Mason is filling a cylindrical water tank that has a radius of 2 feet and a height of 7 feet. He fills it at a constant rate of 2 cubic feet per minute. After 22 minutes, he stops for 7 minutes, then continues filling at the same rate until the tank is full. Sketch a qualitative graph of the water height in the tank versus time. Describe the shape of the graph and explain any changes in slope. Answer: The graph is a piecewise linear function: increasing with a positive slope from 0 to 22 minutes, horizontal (slope 0) from 22 to 29 minutes, then increasing with the same positive slope from 29 minutes until the tank is full at approximately 51.5 minutes. Solution: Calculate the total volume of the tank. Volume = π × r^2 × h = π × (2)^2 × 7 = π × 4 × 7 = 28π ≈ 87.96 cubic feet. Determine the filling rate.
   Full step-by-step solution

   Step 1: Calculate the total volume of the tank. Volume = π × r^2 × h = π × (2)^2 × 7 = π × 4 × 7 = 28π ≈ 87.96 cubic feet. Step 2: Determine the filling rate. The rate is 2 cubic feet per minute. Since the radius is constant, the height increases at a constant rate when filling. The rate of height increase = rate of volume increase / (π × r^2) = 2 / (π × 4) = 2 / (4π) = 1/(2π) feet per minute. Step 3: From 0 to 22 minutes, water is added at constant rate, so height increases linearly with slope 1/(2π). Step 4: From 22 to 29 minutes, no water is added, so height remains constant (slope 0). Step 5: From 29 minutes onward, filling resumes at same rate, so height increases linearly again with slope 1/(2π). Step 6: Find when tank is full. Volume added in first 22 minutes = 2 × 22 = 44 cubic feet. Remaining volume = 28π - 44 ≈ 87.96 - 44 = 43.96 cubic feet. Time needed = 43.96 / 2 ≈ 21.98 minutes. Total time = 29 + 21.98 = 50.98 minutes, approximately 51.5 minutes. Step 7: The graph is a piecewise linear function: increasing from (0,0) to (22, h1), constant from (22, h1) to (29, h1), then increasing from (29, h1) to (51.5, 7). The slope during filling periods is positive and constant; the slope during the break is zero. The answer is as described.

3. (4.8 × 10^6) ÷ (1.2 × 10^3) = ? Answer: 4000 Solution: Separate the decimal numbers and powers of 10: (4.8 ÷ 1.2) × (10^6 ÷ 10^3) Divide the decimal numbers: 4.8 ÷ 1.2 = 4 Divide the powers of 10: 10^6 ÷ 10^3 = 10^(6-3) = 10^3 Multiply the results: 4 × 10^3 = 4 × 1000 = 4000 The answer is 4000.
   Full step-by-step solution

   Step 1: Separate the decimal numbers and powers of 10: (4.8 ÷ 1.2) × (10^6 ÷ 10^3) Step 2: Divide the decimal numbers: 4.8 ÷ 1.2 = 4 Step 3: Divide the powers of 10: 10^6 ÷ 10^3 = 10^(6-3) = 10^3 Step 4: Multiply the results: 4 × 10^3 = 4 × 1000 = 4000 The answer is 4000.

4. Noah is filling a rectangular prism-shaped tank with water. The tank has a base area of 6 square meters and a height of 4 meters. Water flows into the tank at a constant rate of 2 cubic meters per minute. Sketch a qualitative graph of the water volume in the tank versus time, from the moment the tank is empty until it is full. Label the axes and indicate the time when the tank is full. Answer: Graph: Volume (y-axis) vs Time (x-axis). A straight line from (0,0) to (12,24). Tank full at 12 minutes. Solution: Find the total volume of the tank. Volume = base area × height = 6 × 4 = 24 cubic meters. The water flows in at 2 cubic meters per minute.
   Full step-by-step solution

   Step 1: Find the total volume of the tank. Volume = base area × height = 6 × 4 = 24 cubic meters. Step 2: The water flows in at 2 cubic meters per minute. Since the flow rate is constant, the volume increases linearly with time. Step 3: Find the time to fill the tank: time = total volume / flow rate = 24 / 2 = 12 minutes. Step 4: Sketch the graph: Draw a horizontal axis labeled 'Time (minutes)' from 0 to 12. Draw a vertical axis labeled 'Volume (cubic meters)' from 0 to 24. Plot a straight line starting at (0,0) and ending at (12,24). The line has a constant slope, showing the volume increases steadily until the tank is full at 12 minutes.

5. Liam is designing a rectangular garden for his school. The length of the garden is 5 meters more than twice its width. If the perimeter of the garden is 70 meters, what is the width of the garden in meters? Answer: 10 Solution: Let the width of the garden be \( w \) meters. The length is 5 meters more than twice the width, so: Length \( l = 2w + 5 \). \( P = 2 \times (\text{length} + \text{width}) \).
   Full step-by-step solution

   Let's solve the problem step by step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 5 meters more than twice the width, so: Length \( l = 2w + 5 \). --- **Step 2: Write the perimeter formula** The perimeter \( P \) of a rectangle is: \( P = 2 \times (\text{length} + \text{width}) \). Given \( P = 70 \), we have: \( 2 \times (l + w) = 70 \). --- **Step 3: Substitute the expression for length** Substitute \( l = 2w + 5 \) into the perimeter equation: \( 2 \times ( (2w + 5) + w ) = 70 \). --- **Step 4: Simplify inside the parentheses** \( (2w + 5) + w = 3w + 5 \). So: \( 2 \times (3w + 5) = 70 \). --- **Step 5: Solve for \( w \)** Divide both sides by 2: \( 3w + 5 = 35 \). Subtract 5 from both sides: \( 3w = 30 \). Divide by 3: \( w = 10 \). --- **Step 6: Conclusion** The width of the garden is 10 meters. --- **Final answer:** 10

6. Emma is organizing a school fundraiser and needs to create a budget. She has $500 to spend on supplies. She buys t-shirts that cost $8 each and water bottles that cost $5 each. If she buys 40 t-shirts, how many water bottles can she buy while spending exactly her $500 budget? Answer: 36 Solution: Step 1: Calculate the cost of the t-shirts: 40 t-shirts × $8 per t-shirt = $320 Step 2: Subtract the t-shirt cost from the total budget: $500 - $320 = $180 remaining Step 3: Divide the remaining money by the cost per water bottle: $180 ÷ $5 per water bottle = 36 water bottles Step 4: Verify: (40…
   Full step-by-step solution

   Step 1: Calculate the cost of the t-shirts: 40 t-shirts × $8 per t-shirt = $320 Step 2: Subtract the t-shirt cost from the total budget: $500 - $320 = $180 remaining Step 3: Divide the remaining money by the cost per water bottle: $180 ÷ $5 per water bottle = 36 water bottles Step 4: Verify: (40 × $8) + (36 × $5) = $320 + $180 = $500 The answer is 36 water bottles.