Linear with Rationals

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Linear with Rationals · Grade 9 · Worksheet 2

1. A robotics team is programming a drone to follow a parabolic path described by the equation h(t) = -2t² + 12t, where h represents height in meters and t represents time in seconds. The team wants to know at what time the drone will reach its maximum height. Find the time when the drone reaches its highest point. Answer: 3 Solution: The height function is h(t) = -2t² + 12t This is a quadratic function in the form at² + bt + c, where a = -2 and b = 12 For a quadratic function, the vertex occurs at t = -b/(2a) Substitute the values: t = -12/(2×(-2)) = -12/(-4) = 3 The drone reaches its maximum height at t = 3 seconds The…
   Full step-by-step solution

   Step 1: The height function is h(t) = -2t² + 12t Step 2: This is a quadratic function in the form at² + bt + c, where a = -2 and b = 12 Step 3: For a quadratic function, the vertex occurs at t = -b/(2a) Step 4: Substitute the values: t = -12/(2×(-2)) = -12/(-4) = 3 Step 5: The drone reaches its maximum height at t = 3 seconds The answer is 3.

2. Hana is mixing a chemical solution for an experiment. She needs to add a certain amount of a concentrated acid to a beaker containing 0.75 liters of water. The final concentration of acid in the solution must be 0.4 (or 40%). If the concentrated acid is 100% pure, how many liters of acid should Hana add to achieve the desired concentration? Represent the situation with an equation and solve. Answer: 0.5 Solution: Let x = liters of pure acid to add. The amount of pure acid is x. The total volume after adding is (x + 0.75) liters.
   Full step-by-step solution

   Step 1: Let x = liters of pure acid to add. The amount of pure acid is x. The total volume after adding is (x + 0.75) liters. The concentration is x / (x + 0.75). Set this equal to 0.4: x / (x + 0.75) = 0.4. Step 2: Multiply both sides by (x + 0.75): x = 0.4(x + 0.75). Step 3: Distribute: x = 0.4x + 0.3. Step 4: Subtract 0.4x from both sides: 0.6x = 0.3. Step 5: Divide by 0.6: x = 0.3 / 0.6 = 0.5. The answer is 0.5 liters.

3. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,3). The hypotenuse has a length of 5 units. Using the Pythagorean theorem, determine the value of x. Answer: 4 Solution: We are given a right triangle with vertices at (0,0), (x,0), and (0,3). The hypotenuse has length 5 units. Identify the legs of the triangle.
   Full step-by-step solution

   We are given a right triangle with vertices at (0,0), (x,0), and (0,3). The hypotenuse has length 5 units. Step 1: Identify the legs of the triangle. The point (x,0) is on the x-axis, so the distance from (0,0) to (x,0) is |x|. The point (0,3) is on the y-axis, so the distance from (0,0) to (0,3) is 3. Thus, the two legs of the right triangle have lengths |x| and 3. Step 2: Apply the Pythagorean theorem. The hypotenuse is 5, so: (leg1)^2 + (leg2)^2 = (hypotenuse)^2 |x|^2 + 3^2 = 5^2 Step 3: Simplify. x^2 + 9 = 25 Step 4: Solve for x^2. x^2 = 25 - 9 x^2 = 16 Step 5: Solve for x. x = 4 or x = -4 Step 6: Interpret the result. Since the vertex (x,0) is to the right of (0,0) in a typical drawing, x > 0, so x = 4. Final answer: 4

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (2x,0), and (0,x). The area of the triangle is 16 square units. Find the value of x. Answer: 4 Solution: Identify the base and height of the triangle. The base is along the x-axis from (0,0) to (2x,0), so its length is 2x. The height is along the y-axis from (0,0) to (0,x), so its length is x.
   Full step-by-step solution

   Step 1: Identify the base and height of the triangle. The base is along the x-axis from (0,0) to (2x,0), so its length is 2x. The height is along the y-axis from (0,0) to (0,x), so its length is x. Step 2: Use the area formula for a triangle: Area = (1/2) × base × height. Step 3: Substitute the known values: 16 = (1/2) × (2x) × (x). Step 4: Simplify the equation: 16 = (1/2) × 2x × x = (1/2) × 2x² = x². Step 5: Solve for x: x² = 16, so x = 4 (since length must be positive). The answer is 4.

5. Noah is calculating the total resistance in an electrical circuit with two resistors connected in parallel. The total resistance R (in ohms) satisfies the equation 2/3R + 1/4 = 7/12. Solve for R to find the total resistance in ohms. Answer: 0.5 Solution: Start with the equation 2/3R + 1/4 = 7/12. Subtract 1/4 from both sides: 2/3R = 7/12 - 1/4. Convert 1/4 to twelfths: 1/4 = 3/12.
   Full step-by-step solution

   Step 1: Start with the equation 2/3R + 1/4 = 7/12. Step 2: Subtract 1/4 from both sides: 2/3R = 7/12 - 1/4. Step 3: Convert 1/4 to twelfths: 1/4 = 3/12. Step 4: Subtract: 7/12 - 3/12 = 4/12 = 1/3. Step 5: Now the equation is 2/3R = 1/3. Step 6: Multiply both sides by the reciprocal of 2/3, which is 3/2: R = (1/3) * (3/2) = 1/2. Step 7: R = 1/2 ohm = 0.5 ohm. The answer is 0.5.

6. A chemical engineer is designing a reactor where the concentration of a reactant decreases over time according to the function C(t) = (2t² - 18) / (3t - 9), where t represents time in hours and t > 3. At what time does the concentration reach zero? Answer: 3 Solution: Step 1: Set the concentration function equal to zero: (2t² - 18) / (3t - 9) = 0 Step 2: A fraction equals zero when its numerator equals zero (and denominator is not zero) Step 3: Set numerator equal to zero: 2t² - 18 = 0 Step 4: Add 18 to both sides: 2t² = 18 Step 5: Divide both sides by 2: t²…
   Full step-by-step solution

   Step 1: Set the concentration function equal to zero: (2t² - 18) / (3t - 9) = 0 Step 2: A fraction equals zero when its numerator equals zero (and denominator is not zero) Step 3: Set numerator equal to zero: 2t² - 18 = 0 Step 4: Add 18 to both sides: 2t² = 18 Step 5: Divide both sides by 2: t² = 9 Step 6: Take square root of both sides: t = 3 or t = -3 Step 7: Since t > 3 (given in the problem), and t = -3 is not valid for time, the only valid solution is t = 3 Step 8: Verify denominator is not zero when t = 3: 3(3) - 9 = 9 - 9 = 0 Step 9: Since the denominator equals zero when t = 3, this value is excluded from the domain Step 10: Therefore, there is no time t > 3 where the concentration reaches zero The answer is that there is no solution within the given domain.

7. Aroha is calculating the total cost of a school trip. The bus rental fee is $250, and the cost per student for food and entry tickets is represented by x dollars. There are 32 students going on the trip. The total cost for the trip is $842. If the equation representing the total cost is 32x + 250 = 842, what is the cost per student (x) in dollars? Answer: 18.50 Solution: The equation is 32x + 250 = 842. Step 2: Subtract 250 from both sides: 32x = 842 - 250, so 32x = 592. Step 3: Divide both sides by 32: x = 592 / 32.
   Full step-by-step solution

   Step 1: The equation is 32x + 250 = 842. Step 2: Subtract 250 from both sides: 32x = 842 - 250, so 32x = 592. Step 3: Divide both sides by 32: x = 592 / 32. Step 4: Simplify: x = 18.5. The cost per student is $18.50.

8. Aroha draws a rectangle on a coordinate plane with vertices at (0,0), (3x,0), (3x,2/3 x), and (0,2/3 x). The area of the rectangle is 72 square units. Find the value of x. Answer: 6 Solution: Identify the side lengths from the coordinates. The rectangle extends from x=0 to x=3x, so the length is 3x. It extends from y=0 to y=2/3 x, so the width is 2/3 x.
   Full step-by-step solution

   Step 1: Identify the side lengths from the coordinates. The rectangle extends from x=0 to x=3x, so the length is 3x. It extends from y=0 to y=2/3 x, so the width is 2/3 x. Step 2: Write the area formula for a rectangle: Area = length x width. Step 3: Substitute the given area and expressions: 72 = (3x) x (2/3 x). Step 4: Multiply the terms: 3x x 2/3 x = (3 x 2/3) x (x x x) = 2 x^2. So the equation is 72 = 2x^2. Step 5: Solve for x^2: x^2 = 72 / 2 = 36. Step 6: Solve for x: x = sqrt(36) = 6 (since length is positive). The answer is 6.