Rational Equations

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Rational Equations · Grade 9 · Worksheet 2

1. A chemical reaction proceeds at a rate modeled by the function r(t) = (2t^2 - 8)/(t - 2) grams per minute, where t is time in minutes. Maya needs to determine the instantaneous rate of the reaction at t = 4 minutes, but the function appears undefined at t = 2. By simplifying the rational expression, what is the actual rate of the reaction at t = 4 minutes? Answer: 12 Solution: Start with the given function: r(t) = (2t^2 - 8)/(t - 2) Factor the numerator: 2t^2 - 8 = 2(t^2 - 4) = 2(t - 2)(t + 2) Rewrite the function: r(t) = [2(t - 2)(t + 2)]/(t - 2) Cancel the common factor (t - 2): r(t) = 2(t + 2) for t ≠ 2 Evaluate at t = 4: r(4) = 2(4 + 2) = 2(6) = 12 The rate of the…
   Full step-by-step solution

   Step 1: Start with the given function: r(t) = (2t^2 - 8)/(t - 2) Step 2: Factor the numerator: 2t^2 - 8 = 2(t^2 - 4) = 2(t - 2)(t + 2) Step 3: Rewrite the function: r(t) = [2(t - 2)(t + 2)]/(t - 2) Step 4: Cancel the common factor (t - 2): r(t) = 2(t + 2) for t ≠ 2 Step 5: Evaluate at t = 4: r(4) = 2(4 + 2) = 2(6) = 12 Step 6: The rate of the reaction at t = 4 minutes is 12 grams per minute.

2. (3x + 2)/(x - 1) = 4 = ? Answer: 6 Solution: Start with the equation (3x + 2)/(x - 1) = 4 Multiply both sides by (x - 1) to eliminate the denominator: 3x + 2 = 4(x - 1) Distribute the 4 on the right side: 3x + 2 = 4x - 4 Subtract 3x from both sides: 2 = x - 4 Add 4 to both sides: 6 = x Check that x = 6 doesn't make the denominator zero (6…
   Full step-by-step solution

   Step 1: Start with the equation (3x + 2)/(x - 1) = 4 Step 2: Multiply both sides by (x - 1) to eliminate the denominator: 3x + 2 = 4(x - 1) Step 3: Distribute the 4 on the right side: 3x + 2 = 4x - 4 Step 4: Subtract 3x from both sides: 2 = x - 4 Step 5: Add 4 to both sides: 6 = x Step 6: Check that x = 6 doesn't make the denominator zero (6 - 1 = 5 ≠ 0) The solution is x = 6.

3. A chemical reaction follows the rate equation R(t) = (2t^2 - 8)/(t^2 - 4), where R is the reaction rate in moles per minute and t is time in minutes. At what time does the reaction rate become undefined due to a vertical asymptote? Answer: 2 Solution: Step 1: Identify when the reaction rate is undefined by setting the denominator equal to zero: t^2 - 4 = 0 Step 2: Solve for t: t^2 = 4, so t = 2 or t = -2 Step 3: Since time cannot be negative in this context, we only consider t = 2 Step 4: Verify this is a vertical asymptote by checking if the…
   Full step-by-step solution

   Step 1: Identify when the reaction rate is undefined by setting the denominator equal to zero: t^2 - 4 = 0 Step 2: Solve for t: t^2 = 4, so t = 2 or t = -2 Step 3: Since time cannot be negative in this context, we only consider t = 2 Step 4: Verify this is a vertical asymptote by checking if the numerator is also zero at t = 2: 2(2)^2 - 8 = 2(4) - 8 = 8 - 8 = 0 Step 5: Since both numerator and denominator equal zero at t = 2, we need to simplify the function: R(t) = (2t^2 - 8)/(t^2 - 4) = 2(t^2 - 4)/(t^2 - 4) = 2 Step 6: The simplified function shows there is actually a hole at t = 2, not a vertical asymptote Step 7: Let's reconsider: The denominator t^2 - 4 = (t - 2)(t + 2) has zeros at t = 2 and t = -2 Step 8: At t = -2, the numerator is 2(-2)^2 - 8 = 2(4) - 8 = 8 - 8 = 0 Step 9: At t = 2, the numerator is also 0 as shown in step 4 Step 10: Since both numerator and denominator are zero at both t = 2 and t = -2, the function simplifies to R(t) = 2 with holes at t = 2 and t = -2 Step 11: Therefore, there are no vertical asymptotes in this function Step 12: Let me correct my approach: The problem asks when the reaction rate becomes undefined, which occurs when the denominator equals zero: t^2 - 4 = 0 Step 13: Solving t^2 - 4 = 0 gives t = 2 or t = -2 Step 14: Since time cannot be negative, the reaction rate becomes undefined at t = 2 minutes Step 15: Even though this is a removable discontinuity (hole), the function is still undefined at this point The answer is 2.

4. Noah is baking cookies for a school bake sale. The recipe calls for 1 cups of flour to make 5 dozen cookies. Noah wants to make 15 dozen cookies. How many cups of flour are needed? Answer: 3 Solution: The ratio is 1 cups per 5 dozen. Step 2: Set up proportion: cups / 15 = 1 / 5. Step 3: Cross-multiply: cups * 5 = 1 * 15.
   Full step-by-step solution

   Step 1: The ratio is 1 cups per 5 dozen. Step 2: Set up proportion: cups / 15 = 1 / 5. Step 3: Cross-multiply: cups * 5 = 1 * 15. Step 4: cups = (1 * 15) / 5. Step 5: cups = 3. The answer is 3 cups.

5. Mere is analyzing a geometric pattern on a coordinate grid. A line segment from P(0,0) to Q(15,0) forms the base of a triangle. A third point R lies directly above the base such that the ratio of the vertical height to the horizontal distance from P to the foot of the altitude is expressed by the equation (x+12)/5 = 3. Find the x-coordinate of the foot of the altitude from R onto the base. Answer: 3 Solution: We are given the equation (x+12)/5 = 3. Step 2: Multiply both sides by 5 to clear the denominator: 5 * (x+12)/5 = 3 * 5, which simplifies to x+12 = 15. Step 3: Subtract 12 from both sides: x+12-12 = 15-12, so x = 3.
   Full step-by-step solution

   Step 1: We are given the equation (x+12)/5 = 3. Step 2: Multiply both sides by 5 to clear the denominator: 5 * (x+12)/5 = 3 * 5, which simplifies to x+12 = 15. Step 3: Subtract 12 from both sides: x+12-12 = 15-12, so x = 3. Step 4: The x-coordinate of the foot of the altitude is 3. The answer is 3.

6. 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.

7. A chemical reaction proceeds at a rate inversely proportional to the concentration of a catalyst. When the catalyst concentration is 0.4 mol/L, the reaction rate is 15 mol/min. The lab technician needs to achieve a reaction rate of 10 mol/min. What catalyst concentration should she use? Answer: 0.6 Solution: Set up the inverse proportion relationship: rate × concentration = k (constant) Use the given values to find k: 15 × 0.4 = 6 So the equation is: rate × concentration = 6 Plug in the desired rate of 10: 10 × concentration = 6 Solve for concentration: concentration = 6 ÷ 10 = 0.6 The required…
   Full step-by-step solution

   Step 1: Set up the inverse proportion relationship: rate × concentration = k (constant) Step 2: Use the given values to find k: 15 × 0.4 = 6 Step 3: So the equation is: rate × concentration = 6 Step 4: Plug in the desired rate of 10: 10 × concentration = 6 Step 5: Solve for concentration: concentration = 6 ÷ 10 = 0.6 Step 6: The required catalyst concentration is 0.6 mol/L