Linear with Rationals

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Answer Key & Explanations

Linear with Rationals · Grade 9 · Worksheet 3

1. Aroha is mixing a chemical solution for an experiment. She needs to create 15 liters of a mixture that is 35% acid. She has two solutions available: one that is 20% acid and another that is 50% acid. How many liters of the 50% acid solution should she use? Answer: 7.5 Solution: Let x be the number of liters of the 50% acid solution. Then the remaining (15 - x) liters come from the 20% acid solution. The amount of pure acid in the 50% solution is 0.50x.
   Full step-by-step solution

   Step 1: Let x be the number of liters of the 50% acid solution. Then the remaining (15 - x) liters come from the 20% acid solution. Step 2: The amount of pure acid in the 50% solution is 0.50x. The amount in the 20% solution is 0.20(15 - x). The total acid in the final mixture is 0.35 * 15 = 5.25 liters. Step 3: Write the equation: 0.50x + 0.20(15 - x) = 5.25 Step 4: Expand: 0.50x + 3 - 0.20x = 5.25 Step 5: Combine like terms: 0.30x + 3 = 5.25 Step 6: Subtract 3 from both sides: 0.30x = 2.25 Step 7: Divide both sides by 0.30: x = 2.25 / 0.30 = 7.5 The answer is 7.5.

2. Liam is designing a rectangular garden with a length that is 3 meters more than twice its width. The area of the garden must be exactly 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 \( l = 2w + 3 \). Area of a rectangle = length × width.
   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 a rectangle = length × width. Area = \( l \times w = (2w + 3) \times w \). Given area = 65 m², so: \[ w(2w + 3) = 65 \] --- **Step 3: Expand and rearrange** \[ 2w^2 + 3w = 65 \] \[ 2w^2 + 3w - 65 = 0 \] --- **Step 4: Solve the quadratic equation** We solve \( 2w^2 + 3w - 65 = 0 \) using the quadratic formula: \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 2 \), \( b = 3 \), \( c = -65 \). First, discriminant \( D = b^2 - 4ac \): \[ D = 3^2 - 4(2)(-65) = 9 + 520 = 529 \] \[ \sqrt{D} = \sqrt{529} = 23 \] So: \[ w = \frac{-3 \pm 23}{2 \times 2} = \frac{-3 \pm 23}{4} \] --- **Step 5: Two possible solutions** Case 1: \( w = \frac{-3 + 23}{4} = \frac{20}{4} = 5 \) Case 2: \( w = \frac{-3 - 23}{4} = \frac{-26}{4} = -6.5 \) Width cannot be negative, so \( w = 5 \) m. --- **Step 6: Find length** \[ l = 2w + 3 = 2(5) + 3 = 10 + 3 = 13 \ \text{m} \] --- **Step 7: Check area** Area = \( 5 \times 13 = 65 \) m², correct. --- **Final answer:** width = 5 m, length = 13 m

3. 2(x - 3)/5 + (x + 4)/3 = 7 Answer: 11 Solution: Multiply both sides by the least common denominator of 5 and 3, which is 15 15 × [2(x - 3)/5 + (x + 4)/3] = 15 × 7 Distribute the 15 to each term 15 × 2(x - 3)/5 + 15 × (x + 4)/3 = 105 3 × 2(x - 3) + 5 × (x + 4) = 105 6(x - 3) + 5(x + 4) = 105 6x - 18 + 5x + 20 = 105 11x + 2 = 105 Subtract 2…
   Full step-by-step solution

   Step 1: Multiply both sides by the least common denominator of 5 and 3, which is 15 15 × [2(x - 3)/5 + (x + 4)/3] = 15 × 7 Step 2: Distribute the 15 to each term 15 × 2(x - 3)/5 + 15 × (x + 4)/3 = 105 Step 3: Simplify each term 3 × 2(x - 3) + 5 × (x + 4) = 105 6(x - 3) + 5(x + 4) = 105 Step 4: Distribute the coefficients 6x - 18 + 5x + 20 = 105 Step 5: Combine like terms 11x + 2 = 105 Step 6: Subtract 2 from both sides 11x = 103 Step 7: Divide both sides by 11 x = 103/11 x = 11 The answer is 11.

4. A robotics team is programming a drone to follow a parabolic path described by the function h(t) = -2t² + 12t + 8, where h represents height in meters and t represents time in seconds. The team needs to determine when the drone will return to its launch height of 8 meters. At what time (other than t=0) does the drone reach 8 meters again? Answer: 6 Solution: Set up the equation using the given height function and launch height -2t² + 12t + 8 = 8 Subtract 8 from both sides to simplify -2t² + 12t = 0 -2t(t - 6) = 0 -2t = 0 or t - 6 = 0 t = 0 or t = 6 The drone returns to 8 meters at t = 6 seconds.
   Full step-by-step solution

   Step 1: Set up the equation using the given height function and launch height -2t² + 12t + 8 = 8 Step 2: Subtract 8 from both sides to simplify -2t² + 12t = 0 Step 3: Factor out the common term -2t(t - 6) = 0 Step 4: Set each factor equal to zero -2t = 0 or t - 6 = 0 Step 5: Solve each equation t = 0 or t = 6 Step 6: Identify the non-zero solution The drone returns to 8 meters at t = 6 seconds. The answer is 6.

5. A physics class is studying projectile motion. The height h (in meters) of a ball thrown upward is given by h(t) = -5t² + 20t + 1.5, where t is time in seconds. At what time does the ball reach its maximum height? Answer: 2 Solution: The height function is h(t) = -5t² + 20t + 1.5 This is a quadratic function in the form at² + bt + c, where a = -5, b = 20, c = 1.5 For a quadratic function, the maximum occurs at t = -b/(2a) Substitute the values: t = -20/(2×(-5)) Calculate: t = -20/(-10) Simplify: t = 2 The ball reaches its…
   Full step-by-step solution

   Step 1: The height function is h(t) = -5t² + 20t + 1.5 Step 2: This is a quadratic function in the form at² + bt + c, where a = -5, b = 20, c = 1.5 Step 3: For a quadratic function, the maximum occurs at t = -b/(2a) Step 4: Substitute the values: t = -20/(2×(-5)) Step 5: Calculate: t = -20/(-10) Step 6: Simplify: t = 2 Step 7: The ball reaches its maximum height at t = 2 seconds

6. A robotics team is programming a robot to follow a parabolic path. The robot's height above ground is modeled by the function h(t) = -2t² + 12t + 8, where h is height in meters and t is time in seconds. At what time will the robot reach its maximum height? Answer: 3 Solution: The function h(t) = -2t² + 12t + 8 is a quadratic function in the form at² + bt + c, where a = -2, b = 12, and c = 8. For a quadratic function, the maximum or minimum occurs at t = -b/(2a).
   Full step-by-step solution

   Step 1: The function h(t) = -2t² + 12t + 8 is a quadratic function in the form at² + bt + c, where a = -2, b = 12, and c = 8. Step 2: For a quadratic function, the maximum or minimum occurs at t = -b/(2a). Step 3: Substitute the values: t = -12/(2 × -2) = -12/(-4) = 3. Step 4: The robot reaches its maximum height at t = 3 seconds. The answer is 3.

7. Matiu is mixing a chemical solution for an experiment. He needs to create a total of 20 liters of a mixture that contains a specific concentration of acid. He has two tanks: one contains a 30% acid solution and the other contains a 60% acid solution. If he uses x liters from the 60% tank and the rest from the 30% tank, the amount of pure acid in the final mixture is given by the expression 0.6x + 0.3(20 - x). Matiu wants the final mixture to contain exactly 9 liters of pure acid. How many liters should he take from the 60% acid tank? Answer: 10 Solution: Set up the equation: 0.6x + 0.3(20 - x) = 9. Distribute 0.3: 0.6x + 6 - 0.3x = 9. Combine like terms: (0.6x - 0.3x) + 6 = 9, so 0.3x + 6 = 9.
   Full step-by-step solution

   Step 1: Set up the equation: 0.6x + 0.3(20 - x) = 9. Step 2: Distribute 0.3: 0.6x + 6 - 0.3x = 9. Step 3: Combine like terms: (0.6x - 0.3x) + 6 = 9, so 0.3x + 6 = 9. Step 4: Subtract 6 from both sides: 0.3x = 3. Step 5: Divide both sides by 0.3: x = 3 / 0.3 = 10. The answer is 10.