Rational Equations

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Rational Equations · Grade 9 · Worksheet 3

1. Mere is planning a road trip. The distance to the destination is 455 kilometers. If Mere drives at a constant speed, the time taken, in hours, is given by the equation 455/v = 5, where v is the speed in kilometers per hour. What is Mere's driving speed? Answer: 91 Solution: The equation is 455/v = 5. Multiply both sides by v: 455 = 5v. Divide both sides by 5: v = 455/5 = 91.
   Full step-by-step solution

   Step 1: The equation is 455/v = 5. Step 2: Multiply both sides by v: 455 = 5v. Step 3: Divide both sides by 5: v = 455/5 = 91. Mere drives at 91 kilometers per hour.

2. A chemical reaction produces a compound at a rate modeled by the function R(t) = (3t^2 - 12)/(t - 2), where t is time in minutes and R(t) is the production rate in grams per minute. Maya needs to determine the instantaneous production rate at t = 2 minutes, but the function appears undefined at this point. What is the actual production rate at t = 2 minutes? Answer: 12 Solution: Factor the numerator: 3t^2 - 12 = 3(t^2 - 4) = 3(t - 2)(t + 2) Rewrite the function: R(t) = [3(t - 2)(t + 2)]/(t - 2) Cancel the common factor (t - 2) from numerator and denominator: R(t) = 3(t + 2) for t ≠ 2 Evaluate the simplified function at t = 2: R(2) = 3(2 + 2) = 3 × 4 = 12 The production…
   Full step-by-step solution

   Step 1: Factor the numerator: 3t^2 - 12 = 3(t^2 - 4) = 3(t - 2)(t + 2) Step 2: Rewrite the function: R(t) = [3(t - 2)(t + 2)]/(t - 2) Step 3: Cancel the common factor (t - 2) from numerator and denominator: R(t) = 3(t + 2) for t ≠ 2 Step 4: Evaluate the simplified function at t = 2: R(2) = 3(2 + 2) = 3 × 4 = 12 Step 5: The production rate at t = 2 minutes is 12 grams per minute.

3. On a coordinate plane, a rectangle has vertices at A(0,0), B(12,0), C(12,8), and D(0,8). A diagonal is drawn from A to C. A point P lies on diagonal AC such that the ratio of the length of AP to the length of PC is 3:5. What is the x-coordinate of point P? Answer: 4.5 Solution: Identify the coordinates of A(0,0) and C(12,8). The ratio AP:PC = 3:5 means AP is 3 parts out of the total 8 parts of AC.
   Full step-by-step solution

   Step 1: Identify the coordinates of A(0,0) and C(12,8). Step 2: The ratio AP:PC = 3:5 means AP is 3 parts out of the total 8 parts of AC. Step 3: The x-coordinate of P is found by starting at A's x-coordinate and adding 3/8 of the horizontal distance from A to C. Step 4: Horizontal distance from A to C = 12 - 0 = 12. Step 5: 3/8 of 12 = (3/8) * 12 = 36/8 = 9/2 = 4.5. Step 6: x-coordinate of P = 0 + 4.5 = 4.5. The answer is 4.5.

4. A chemical reaction proceeds at a rate modeled by the function R(t) = (3t^2 - 12)/(t^2 - 4), where t is time in minutes. Maya needs to determine when the reaction rate becomes undefined due to a discontinuity. At what time value does this occur? Answer: 2 Solution: Identify when the function is undefined by setting the denominator equal to zero: t^2 - 4 = 0 Solve for t: t^2 = 4 t = 2 or t = -2 Since time cannot be negative in this context, we discard t = -2 The reaction rate becomes undefined at t = 2 minutes The answer is 2.
   Full step-by-step solution

   Step 1: Identify when the function is undefined by setting the denominator equal to zero: t^2 - 4 = 0 Step 2: Solve for t: t^2 = 4 Step 3: t = 2 or t = -2 Step 4: Since time cannot be negative in this context, we discard t = -2 Step 5: The reaction rate becomes undefined at t = 2 minutes The answer is 2.

5. Mason is analyzing a geometric pattern on a coordinate grid. A right triangle is drawn with vertices at A(0,0), B(12,0), and C(0,9). A rectangle is inscribed inside the triangle such that one side lies along the x-axis from (0,0) to (x,0) and the opposite side is parallel to the x-axis, touching the hypotenuse at a point. The height of the rectangle is given by the expression (9 - (3/4)x). If the area of the rectangle is expressed as A = x(9 - (3/4)x), and the maximum area occurs when the derivative is zero, find the value of x that maximizes the area. Answer: 6 Solution: Identify the line of the hypotenuse. The triangle has vertices at (0,0), (12,0), and (0,9). The hypotenuse connects (12,0) to (0,9).
   Full step-by-step solution

   Step 1: Identify the line of the hypotenuse. The triangle has vertices at (0,0), (12,0), and (0,9). The hypotenuse connects (12,0) to (0,9). Its slope is (9 - 0)/(0 - 12) = 9/(-12) = -3/4. So the equation of the hypotenuse is y = (-3/4)x + 9. Step 2: The height of the rectangle at any x is the y-coordinate on the hypotenuse, which is y = (-3/4)x + 9. This matches the given expression 9 - (3/4)x. Step 3: The area of the rectangle is width times height: A = x * (9 - (3/4)x) = 9x - (3/4)x^2. Step 4: To find the x that maximizes the area, take the derivative of A with respect to x: dA/dx = 9 - (3/2)x. Step 5: Set the derivative equal to zero: 9 - (3/2)x = 0. Solve for x: (3/2)x = 9, so x = 9 * (2/3) = 6. Step 6: Verify this gives a maximum (second derivative is negative). The value of x that maximizes the area is 6. The answer is 6.

6. Liam is designing a rectangular garden with an area of 48 square meters. The length of the garden is 4 meters more than its width. Write and solve a rational equation to determine the dimensions of Liam's garden. Answer: width = 6 meters, length = 10 meters Solution: In geometry problems involving area and relationships between dimensions, we often use variables to represent the unknown quantities. The area of a rectangle is found by multiplying length and width.
   Full step-by-step solution

   In geometry problems involving area and relationships between dimensions, we often use variables to represent the unknown quantities. The area of a rectangle is found by multiplying length and width. When one dimension is described relative to the other, we can create an equation that incorporates this relationship and the given area. Solving such equations typically involves algebraic manipulation and may require checking for extraneous solutions that don't make sense in the physical context.

7. Mere is analyzing a geometric pattern on a coordinate grid. A rectangle has vertices at A(-6, 0), B(6, 0), C(6, 4), and D(-6, 4). A diagonal is drawn from vertex A to vertex C. A rational equation models the slope of a line parallel to this diagonal: (x - 10) / 2 = 6 / (x + 2). Solve for the value(s) of x. Answer: x = 8, x = -4 Solution: Start with the given equation: (x - 10) / 2 = 6 / (x + 2). Cross-multiply to eliminate the fractions: (x - 10)(x + 2) = 2 * 6. Simplify both sides: (x - 10)(x + 2) = 12.
   Full step-by-step solution

   Step 1: Start with the given equation: (x - 10) / 2 = 6 / (x + 2). Step 2: Cross-multiply to eliminate the fractions: (x - 10)(x + 2) = 2 * 6. Step 3: Simplify both sides: (x - 10)(x + 2) = 12. Expand the left side: x^2 + 2x - 10x - 20 = 12, which simplifies to x^2 - 8x - 20 = 12. Step 4: Subtract 12 from both sides to set the equation to zero: x^2 - 8x - 32 = 0. Step 5: Factor the quadratic: (x - 8)(x + 4) = 0. Step 6: Set each factor equal to zero: x - 8 = 0 gives x = 8; x + 4 = 0 gives x = -4. Step 7: Check for extraneous solutions. The original equation has denominators of 2 and (x + 2). Neither solution makes x + 2 = 0 (since 8 + 2 = 10 and -4 + 2 = -2, neither is zero), so both are valid. The answer is x = 8, x = -4.