Sketch Function Graphs

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Sketch Function Graphs · Grade 8 · Worksheet 3

1. A rectangular prism is drawn with dimensions 8 cm by 5 cm by 3 cm. A diagonal is drawn from the bottom-front-left corner to the top-back-right corner, passing through the interior of the prism. What is the length of this space diagonal? Round your answer to the nearest tenth of a centimeter. Answer: 9.9 Solution: To find the space diagonal of a rectangular prism, use the formula: diagonal = sqrt(length² + width² + height²) Substitute the given dimensions: length = 8 cm, width = 5 cm, height = 3 cm Calculate: diagonal = sqrt(8² + 5² + 3²) = sqrt(64 + 25 + 9) = sqrt(98) sqrt(98) ≈ 9.8995 Round to the…
   Full step-by-step solution

   Step 1: To find the space diagonal of a rectangular prism, use the formula: diagonal = sqrt(length² + width² + height²) Step 2: Substitute the given dimensions: length = 8 cm, width = 5 cm, height = 3 cm Step 3: Calculate: diagonal = sqrt(8² + 5² + 3²) = sqrt(64 + 25 + 9) = sqrt(98) Step 4: sqrt(98) ≈ 9.8995 Step 5: Round to the nearest tenth: 9.9 cm The answer is 9.9.

2. Liam is designing a rectangular garden with a length that is 3 meters more than twice its width. The area of the garden needs to be 65 square meters. What are the dimensions of Liam's garden? Answer: width = 5 m, length = 13 m Solution: Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width, so: length \( l = 2w + 3 \). --- **Step 2: Write the area equation** Area of rectangle = length × width Given area = 65 m², so: \[ (2w + 3) \times w = 65 \] --- **Step 3: Expand and rearrange** \[ 2w^2 + 3w = 65 \] Subtract 65 from both sides: \[ 2w^2 + 3w - 65 = 0 \] --- **Step 4: Solve the quadratic equation** We can use the quadratic formula: For \( a w^2 + b w + c = 0 \), \( a = 2 \), \( b = 3 \), \( c = -65 \). \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] \[ w = \frac{-3 \pm \sqrt{3^2 - 4(2)(-65)}}{2 \times 2} \] \[ w = \frac{-3 \pm \sqrt{9 + 520}}{4} \] \[ w = \frac{-3 \pm \sqrt{529}}{4} \] \[ w = \frac{-3 \pm 23}{4} \] --- **Step 5: Evaluate the two possible solutions** First solution: \[ w = \frac{-3 + 23}{4} = \frac{20}{4} = 5 \] Second solution: \[ w = \frac{-3 - 23}{4} = \frac{-26}{4} = -6.5 \] Since width can't be negative, we take \( w = 5 \). --- **Step 6: Find the length** \[ l = 2w + 3 = 2(5) + 3 = 10 + 3 = 13 \] --- **Step 7: Check the area** Area = \( 13 \times 5 = 65 \) m², which matches the problem. --- **Final answer:** Width = 5 m, Length = 13 m

3. A rectangular prism is drawn with dimensions: length = 8 cm, width = 5 cm, and height = 3 cm. A diagonal is drawn from the bottom-front-left corner to the top-back-right corner, passing through the interior of the prism. What is the length of this space diagonal? Round your answer to the nearest tenth. Answer: 9.9 Solution: The space diagonal of a rectangular prism can be found using the formula: diagonal = sqrt(length^2 + width^2 + height^2). Substitute the given values: length = 8 cm, width = 5 cm, height = 3 cm.
   Full step-by-step solution

   Step 1: The space diagonal of a rectangular prism can be found using the formula: diagonal = sqrt(length^2 + width^2 + height^2). Step 2: Substitute the given values: length = 8 cm, width = 5 cm, height = 3 cm. Step 3: Calculate length^2 = 8^2 = 64. Step 4: Calculate width^2 = 5^2 = 25. Step 5: Calculate height^2 = 3^2 = 9. Step 6: Add the squares: 64 + 25 + 9 = 98. Step 7: Take the square root: sqrt(98) ≈ 9.899. Step 8: Round to the nearest tenth: 9.9 cm. The length of the space diagonal is 9.9 cm.

4. 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.

5. (3.2 × 10^5) ÷ (8 × 10^2) = ? Answer: 400 Solution: Write the division of coefficients: 3.2 ÷ 8 = 0.4 Write the division of powers of 10: 10^5 ÷ 10^2 = 10^(5-2) = 10^3 Combine the results: 0.4 × 10^3 Convert to standard form: 0.4 × 1000 = 400 The answer is 400.
   Full step-by-step solution

   Step 1: Write the division of coefficients: 3.2 ÷ 8 = 0.4 Step 2: Write the division of powers of 10: 10^5 ÷ 10^2 = 10^(5-2) = 10^3 Step 3: Combine the results: 0.4 × 10^3 Step 4: Convert to standard form: 0.4 × 1000 = 400 The answer is 400.