Function End Behavior

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Answer Key & Explanations

Function End Behavior · Grade 12 · Worksheet 2

1. A right circular cone has a height of 12 cm and a base radius of 5 cm. A horizontal plane cuts through the cone parallel to its base, creating a smaller cone at the top and a frustum below. If the smaller cone has a volume exactly one-eighth the volume of the original cone, what is the height of the smaller cone? Answer: 6 cm Solution: Volume = (1/3) * π * r^2 * h Height H = 12 cm Base radius R = 5 cm Volume_original = (1/3) * π * (5)^2 * 12 = (1/3) * π * 25 * 12 = (1/3) * π * 300 = 100π cm³ So Volume_original = 100π Volume_small = (1/8) * Volume_original So Volume_small = (1/8) * 100π = 100π/8 = 12.5π cm³ Relating dimensions…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Volume of the original cone** The formula for the volume of a cone is: Volume = (1/3) * π * r^2 * h For the original cone: Height H = 12 cm Base radius R = 5 cm Volume_original = (1/3) * π * (5)^2 * 12 = (1/3) * π * 25 * 12 = (1/3) * π * 300 = 100π cm³ So Volume_original = 100π --- **Step 2: Volume of the smaller cone** The problem says: Volume_small = (1/8) * Volume_original So Volume_small = (1/8) * 100π = 100π/8 = 12.5π cm³ --- **Step 3: Relating dimensions of small cone to original cone** When a cone is cut parallel to the base, the small top cone is similar to the original cone. Let h = height of the smaller cone (what we want to find). Let r = base radius of the smaller cone. By similarity: r / R = h / H So r / 5 = h / 12 Thus r = (5/12) * h --- **Step 4: Volume of small cone in terms of h** Volume_small = (1/3) * π * r^2 * h = (1/3) * π * [ (5/12) * h ]^2 * h = (1/3) * π * (25/144) * h^2 * h = (1/3) * π * (25/144) * h^3 --- **Step 5: Set equal to known volume** We know Volume_small = 12.5π So: (1/3) * π * (25/144) * h^3 = 12.5π Cancel π from both sides: (1/3) * (25/144) * h^3 = 12.5 Multiply both sides by 3: (25/144) * h^3 = 37.5 Multiply both sides by 144: 25 * h^3 = 37.5 * 144 --- **Step 6: Solve for h^3** 37.5 * 144 = (75/2) * 144 = 75 * 72 = 5400 So: 25 * h^3 = 5400 h^3 = 5400 / 25 = 216 --- **Step 7: Find h** h^3 = 216 h = cube root of 216 = 6 --- **Step 8: Conclusion** The height of the smaller cone is 6 cm. --- **Final answer:** 6 cm

2. lim(x→-∞) (9x⁵ - 7x³ + 5)/(3x⁵ + 2x² - 11) = ? Answer: 3 Solution: Identify the highest degree terms in numerator and denominator. Numerator: 9x⁵ (degree 5). Denominator: 3x⁵ (degree 5).
   Full step-by-step solution

   Step 1: Identify the highest degree terms in numerator and denominator. Numerator: 9x⁵ (degree 5). Denominator: 3x⁵ (degree 5). Step 2: Since the degrees are equal, divide both numerator and denominator by x⁵: (9x⁵/x⁵ - 7x³/x⁵ + 5/x⁵) / (3x⁵/x⁵ + 2x²/x⁵ - 11/x⁵) = (9 - 7/x² + 5/x⁵) / (3 + 2/x³ - 11/x⁵) Step 3: As x→-∞, terms with x in the denominator approach 0: 7/x² → 0, 5/x⁵ → 0, 2/x³ → 0, 11/x⁵ → 0. Step 4: The limit simplifies to (9 - 0 + 0) / (3 + 0 - 0) = 9/3 = 3. The answer is 3.

3. lim(x→-∞) (8x⁶ - 6x⁴ + 2)/(4x⁶ + 10x³ - 2) = ? Answer: 2 Solution: Identify the highest degree terms. Numerator: 8x⁶ (degree 6). Denominator: 4x⁶ (degree 6).
   Full step-by-step solution

   Step 1: Identify the highest degree terms. Numerator: 8x⁶ (degree 6). Denominator: 4x⁶ (degree 6). Step 2: Since degrees are equal, divide both numerator and denominator by x⁶: (8x⁶/x⁶ - 6x⁴/x⁶ + 2/x⁶) / (4x⁶/x⁶ + 10x³/x⁶ - 2/x⁶) = (8 - 6/x² + 2/x⁶) / (4 + 10/x³ - 2/x⁶) Step 3: As x→-∞, terms with x in the denominator approach 0: 6/x² → 0, 2/x⁶ → 0, 10/x³ → 0, -2/x⁶ → 0. Step 4: The limit simplifies to (8 - 0 + 0) / (4 + 0 - 0) = 8/4 = 2. The answer is 2.

4. Aroha is an environmental scientist modeling the decay of a pollutant in a lake. The concentration of the pollutant, C(t) in parts per million, is given by the function C(t) = (5t^3 - 7t^2 + 9) / (t^3 + 3t - 1), where t is time in years. As time goes on indefinitely (t → ∞), what is the limiting concentration of the pollutant in the lake? Answer: 5 Solution: Identify the highest power of t in the numerator and denominator. The numerator is 5t^3 - 7t^2 + 9, so the highest power is t^3 with coefficient 5.
   Full step-by-step solution

   Step 1: Identify the highest power of t in the numerator and denominator. The numerator is 5t^3 - 7t^2 + 9, so the highest power is t^3 with coefficient 5. The denominator is t^3 + 3t - 1, so the highest power is t^3 with coefficient 1. Step 2: For rational functions where the degrees of the numerator and denominator are equal, the limit as t approaches infinity is the ratio of the leading coefficients. Step 3: The leading coefficient of the numerator is 5, and the leading coefficient of the denominator is 1. Step 4: Calculate the ratio: 5 / 1 = 5. Step 5: Therefore, as t → ∞, C(t) approaches 5 parts per million. The answer is 5.

5. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time using the function C(t) = (3t^3 - 2t^2 + 5)/(t^3 - 4), where C is measured in milligrams per liter and t is time in hours. As time approaches infinity, what value does the drug concentration approach, and what does this tell doctors about the long-term behavior of this medication? Answer: 3 mg/L Solution: To find the long-term behavior of the drug concentration as time approaches infinity, we analyze the function: C(t) = (3t^3 - 2t^2 + 5) / (t^3 - 4) Identify the degrees of the numerator and denominator.
   Full step-by-step solution

   To find the long-term behavior of the drug concentration as time approaches infinity, we analyze the function: C(t) = (3t^3 - 2t^2 + 5) / (t^3 - 4) Step 1: Identify the degrees of the numerator and denominator. - The numerator is 3t^3 - 2t^2 + 5. The highest power of t is t^3, so the degree is 3. - The denominator is t^3 - 4. The highest power of t is t^3, so the degree is also 3. Step 2: Since the degrees of the numerator and denominator are equal, the limit as t approaches infinity is determined by the ratio of the leading coefficients. - The leading coefficient of the numerator is 3 (from 3t^3). - The leading coefficient of the denominator is 1 (from t^3). Step 3: Calculate the limit. - The limit as t approaches infinity of C(t) is 3/1 = 3. Step 4: Interpret the result. - This means that as time becomes very large, the drug concentration in the bloodstream approaches 3 mg/L. - This tells doctors that the medication reaches a stable concentration of 3 mg/L in the long term, which may represent a steady-state therapeutic level. Therefore, the drug concentration approaches 3 mg/L.

6. A biomedical engineer is modeling the concentration of a new medication in a patient's bloodstream over time using the function C(t) = (3t^3 - 2t^2 + 5)/(t^3 - 4), where C is measured in milligrams per liter and t is time in hours. As time approaches infinity, what concentration level does the medication approach in the patient's bloodstream? Answer: 3 mg/L Solution: To find the concentration as time approaches infinity, we analyze the function C(t) = (3t^3 - 2t^2 + 5)/(t^3 - 4). Identify the degrees of the numerator and denominator. - The numerator is 3t^3 - 2t^2 + 5.
   Full step-by-step solution

   To find the concentration as time approaches infinity, we analyze the function C(t) = (3t^3 - 2t^2 + 5)/(t^3 - 4). Step 1: Identify the degrees of the numerator and denominator. - The numerator is 3t^3 - 2t^2 + 5. The highest power of t is t^3, so the degree is 3. - The denominator is t^3 - 4. The highest power of t is t^3, so the degree is also 3. Step 2: Since the degrees of the numerator and denominator are equal, the limit as t approaches infinity is determined by the ratio of the leading coefficients. - The leading coefficient of the numerator is 3 (from 3t^3). - The leading coefficient of the denominator is 1 (from t^3). Step 3: Calculate the limit. - The limit as t approaches infinity is (leading coefficient of numerator) / (leading coefficient of denominator) = 3 / 1 = 3. Step 4: Interpret the result. - Therefore, as time approaches infinity, the concentration of the medication in the bloodstream approaches 3 mg/L. Final Answer: 3 mg/L

7. A solid is formed by rotating the region bounded by the curves y = x^2 and y = 4 about the line y = 6. This creates a three-dimensional shape resembling a bowl with a parabolic cross-section. Calculate the volume of this solid of revolution using the method of washers. Answer: 384π/5 Solution: When finding volumes of revolution using washers, we consider the area between two curves rotated around an external axis.
   Full step-by-step solution

   When finding volumes of revolution using washers, we consider the area between two curves rotated around an external axis. The washer method calculates the volume by summing cross-sectional areas perpendicular to the axis of rotation. For regions bounded by curves, we determine the outer and inner radii based on the distance from each curve to the axis of rotation.