Rational Equations

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Rational Equations · Grade 9 · Worksheet 1

1. A chemical reaction follows a rate modeled by the rational function R(t) = (3t^2 - 12) / (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 function 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: Check if this creates a vertical asymptote by verifying the…
   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, so t = 2 or t = -2 Step 3: Since time cannot be negative in this context, we only consider t = 2 Step 4: Check if this creates a vertical asymptote by verifying the numerator isn't zero at t = 2: 3(2)^2 - 12 = 12 - 12 = 0 Step 5: Since both numerator and denominator are zero at t = 2, factor both: R(t) = 3(t^2 - 4)/(t^2 - 4) = 3 for t ≠ 2 Step 6: The function simplifies to a constant value of 3 with a removable discontinuity at t = 2, not a vertical asymptote Step 7: Re-examine the denominator: t^2 - 4 = (t - 2)(t + 2) Step 8: The function has removable discontinuities at both t = 2 and t = -2, meaning there are no vertical asymptotes Step 9: The problem asks when the reaction rate becomes undefined, which occurs at t = 2 (the only positive time value) The answer is 2.

2. Tane is analyzing a geometric pattern on a coordinate grid. A right triangle is drawn with vertices at A(0,0), B(14,0), and C(0,9). Inside this triangle, a rectangle is inscribed such that one side lies along the x-axis from (0,0) to (8,0), and the opposite side touches the hypotenuse of the triangle at points where the rectangle's top-left corner is at (0,5) and top-right corner is at (8,5). What is the area of the rectangular region inside the triangle? Answer: 40 Solution: Identify the width of the rectangle. The rectangle spans from x = 0 to x = 8 along the base, so the width is 8 - 0 = 8 units. Identify the height of the rectangle.
   Full step-by-step solution

   Step 1: Identify the width of the rectangle. The rectangle spans from x = 0 to x = 8 along the base, so the width is 8 - 0 = 8 units. Step 2: Identify the height of the rectangle. The rectangle extends from the x-axis (y = 0) up to y = 5, so the height is 5 - 0 = 5 units. Step 3: Calculate the area of the rectangle using the formula: Area = width * height. Step 4: Area = 8 * 5 = 40 square units. The answer is 40.

3. A chemical reaction follows a rate model where the reaction rate R is given by R(x) = (2x^2 - 5x + 3)/(x - 1), where x represents the concentration of a reactant in moles per liter. At what concentration does the reaction rate reach 7 moles per liter per minute? Answer: 5 Solution: Set up the equation: (2x^2 - 5x + 3)/(x - 1) = 7 Factor the numerator: 2x^2 - 5x + 3 = (2x - 3)(x - 1) Simplify the rational expression: [(2x - 3)(x - 1)]/(x - 1) = 2x - 3, for x ≠ 1 Substitute back into the equation: 2x - 3 = 7 Solve for x: 2x = 10, so x = 5 Check the domain restriction: x = 5…
   Full step-by-step solution

   Step 1: Set up the equation: (2x^2 - 5x + 3)/(x - 1) = 7 Step 2: Factor the numerator: 2x^2 - 5x + 3 = (2x - 3)(x - 1) Step 3: Simplify the rational expression: [(2x - 3)(x - 1)]/(x - 1) = 2x - 3, for x ≠ 1 Step 4: Substitute back into the equation: 2x - 3 = 7 Step 5: Solve for x: 2x = 10, so x = 5 Step 6: Check the domain restriction: x = 5 is not equal to 1, so it is valid. The concentration is 5 moles per liter.

4. (3x - 2)/(x + 4) = 5 = ? Answer: -11 Solution: Multiply both sides by (x + 4) to eliminate the denominator: (3x - 2) = 5(x + 4) Distribute the 5 on the right side: 3x - 2 = 5x + 20 Subtract 3x from both sides: -2 = 2x + 20 Subtract 20 from both sides: -22 = 2x Divide both sides by 2: x = -11 Check that x = -11 doesn't make the original…
   Full step-by-step solution

   Step 1: Multiply both sides by (x + 4) to eliminate the denominator: (3x - 2) = 5(x + 4) Step 2: Distribute the 5 on the right side: 3x - 2 = 5x + 20 Step 3: Subtract 3x from both sides: -2 = 2x + 20 Step 4: Subtract 20 from both sides: -22 = 2x Step 5: Divide both sides by 2: x = -11 Step 6: Check that x = -11 doesn't make the original denominator zero: (-11) + 4 = -7 ≠ 0 The solution is x = -11.

5. A right triangle is inscribed in a circle with diameter 10 units. The triangle's hypotenuse lies along the diameter, and one leg has length 6 units. What is the area of the triangle? Answer: 24 Solution: We have a circle with diameter 10 units. A right triangle is inscribed in the circle, with the hypotenuse along the diameter. One leg of the triangle is 6 units.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the setup** We have a circle with diameter 10 units. A right triangle is inscribed in the circle, with the hypotenuse along the diameter. One leg of the triangle is 6 units. --- **Step 2: Recall a geometry fact** For a right triangle inscribed in a circle, the hypotenuse is the diameter of the circle. So hypotenuse length = 10. --- **Step 3: Apply the Pythagorean theorem** Let the triangle have legs \( a \) and \( b \), and hypotenuse \( c \). We know: \( c = 10 \), \( a = 6 \), \( b = ? \) Pythagorean theorem: \( a^2 + b^2 = c^2 \) Substitute: \( 6^2 + b^2 = 10^2 \) \( 36 + b^2 = 100 \) \( b^2 = 100 - 36 = 64 \) \( b = 8 \) (length is positive) --- **Step 4: Find the area of the triangle** Area = (1/2) × base × height Here, the legs are perpendicular, so: Area = (1/2) × 6 × 8 = (1/2) × 48 = 24 --- **Final answer:** 24

6. 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