Function Parameters

lessonbunny.com

Answer Key & Explanations

Function Parameters · Grade 9 · Worksheet 3

1. Hana is a conservation scientist tracking the population of a rare bird species on a remote island. The number of birds, N(t), can be modeled by the exponential function N(t) = 240(0.85)^t, where t is the number of years since the study began. Interpret the meaning of the parameters 240 and 0.85 in this context. Answer: 240 is the initial number of birds at the start of the study; 0.85 is the decay factor, meaning the population decreases by 15% each year. Solution: The function is N(t) = 240(0.85)^t, which is in the form y = a(b)^x, where a is the initial value and b is the growth or decay factor. The parameter 240 is the value when t = 0.
   Full step-by-step solution

   Step 1: The function is N(t) = 240(0.85)^t, which is in the form y = a(b)^x, where a is the initial value and b is the growth or decay factor. Step 2: The parameter 240 is the value when t = 0. This means at the start of the study (t = 0), there were 240 birds. Step 3: The parameter 0.85 is the base of the exponent. Since 0.85 is less than 1, this indicates exponential decay. Each year, the population is multiplied by 0.85, meaning it retains 85% of the previous year's population. Step 4: The decay rate is 1 - 0.85 = 0.15, or 15% per year. So, the population decreases by 15% each year. Final answer: 240 is the initial number of birds; 0.85 is the decay factor, meaning the population decreases by 15% per year.

2. Noah is analyzing the depreciation of a piece of machinery. Its value, in dollars, after x years is modeled by the exponential function V(x) = 8400(0.85)^x. A graph of this function shows the value decreasing over time, starting from a point on the vertical axis at x=0 and curving downward to approach zero. What is the real-world meaning of the number 8400 in this model, and what does the number 0.85 represent? Answer: 8400 represents the initial value of the machinery in dollars; 0.85 represents the annual decay factor, meaning the machinery retains 85% of its value each year. Solution: The function is V(x) = 8400(0.85)^x. When x = 0, V(0) = 8400(0.85)^0 = 8400(1) = 8400. So 8400 is the initial value of the machinery at the time of purchase.
   Full step-by-step solution

   Step 1: The function is V(x) = 8400(0.85)^x. When x = 0, V(0) = 8400(0.85)^0 = 8400(1) = 8400. So 8400 is the initial value of the machinery at the time of purchase. Step 2: The base 0.85 is less than 1, indicating exponential decay. Since 0.85 = 85/100 = 85%, the machinery retains 85% of its value each year. This is called the decay factor. Step 3: Therefore, 8400 means the machinery was worth $8400 when new, and 0.85 means it loses 15% of its value each year (since 100% - 85% = 15%). Final answer: 8400 represents the initial value in dollars; 0.85 represents the annual decay factor (85% retained per year).

3. Aroha's investment grows according to y = 9500(1.12)^x. What does 9500 represent? What does 1.12 represent? Answer: 9500 represents the initial investment amount, 1.12 represents the growth factor per period (12% increase) Solution: Identify the parameters in the exponential function y = 9500(1.12)^x The parameter 9500 is the coefficient 'a' in the form y = ab^x When x = 0, y = 9500(1.12)^0 = 9500 × 1 = 9500 Therefore, 9500 represents the initial investment amount when time x = 0 The parameter 1.12 is the base 'b' in the…
   Full step-by-step solution

   Step 1: Identify the parameters in the exponential function y = 9500(1.12)^x Step 2: The parameter 9500 is the coefficient 'a' in the form y = ab^x Step 3: When x = 0, y = 9500(1.12)^0 = 9500 × 1 = 9500 Step 4: Therefore, 9500 represents the initial investment amount when time x = 0 Step 5: The parameter 1.12 is the base 'b' in the form y = ab^x Step 6: Each time x increases by 1, the investment multiplies by 1.12 Step 7: This means the investment grows by 12% each period (since 1.12 = 1 + 0.12) Step 8: Therefore, 1.12 represents the growth factor per period, indicating a 12% increase

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). A circle is inscribed in this triangle, tangent to all three sides. What is the radius of the inscribed circle? Answer: 1 Solution: A = (0,0) B = (4,0) C = (4,3) This is a right triangle with the right angle at B = (4,0) because AB is horizontal and BC is vertical.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the triangle** Vertices: A = (0,0) B = (4,0) C = (4,3) This is a right triangle with the right angle at B = (4,0) because AB is horizontal and BC is vertical. Sides: AB = from (0,0) to (4,0) → length = 4 BC = from (4,0) to (4,3) → length = 3 AC = hypotenuse from (0,0) to (4,3) → length = sqrt( (4-0)^2 + (3-0)^2 ) = sqrt(16 + 9) = sqrt(25) = 5 So sides: 3, 4, 5. --- **Step 2: Incenter radius formula for a right triangle** For any triangle, the inradius r = Area / Semiperimeter. Area = (1/2) * base * height Here base = 4, height = 3 (legs of the right triangle). Area = (1/2) * 4 * 3 = 6. Semiperimeter s = (3 + 4 + 5) / 2 = 12 / 2 = 6. --- **Step 3: Compute r** r = Area / s = 6 / 6 = 1. --- **Step 4: Conclusion** The radius of the inscribed circle is 1. --- **Final answer:** 1

5. The graph below (described in text) shows the value of a rare comic book over time. The value V (in dollars) is modeled by the exponential function V(t) = 75(1.05)^t, where t is the number of years since 2010. A coordinate plane is drawn with the horizontal axis labeled 'Years since 2010 (t)' and the vertical axis labeled 'Value (V) in dollars'. The curve starts at (0, 75) and rises slowly, passing through points such as (10, approximately 122) and (20, approximately 199). Interpret the meaning of the parameters 75 and 1.05 in the context of the comic book's value. Answer: 75 is the initial value of the comic book in 2010, and 1.05 is the annual growth factor, meaning the value increases by 5% each year. Solution: Identify the general form of an exponential function: V(t) = a * b^t. Here, a = 75 and b = 1.05. Interpret a.
   Full step-by-step solution

   Step 1: Identify the general form of an exponential function: V(t) = a * b^t. Here, a = 75 and b = 1.05. Step 2: Interpret a. The parameter a represents the initial value when t = 0. Since t is the number of years since 2010, when t = 0, V(0) = 75 * (1.05)^0 = 75 * 1 = 75. So in the year 2010, the comic book was worth $75. Step 3: Interpret b. The parameter b is the growth factor. Since b = 1.05, the value is multiplied by 1.05 each year. A growth factor of 1.05 corresponds to a 5% increase each year (because 1.05 = 1 + 0.05, and 0.05 = 5%). Step 4: Conclusion. The number 75 means the comic book was initially worth $75 in 2010. The number 1.05 means the value grows by a factor of 1.05 each year, which is a 5% annual increase. Final answer: 75 is the initial value in 2010; 1.05 is the annual growth factor (5% increase per year).

6. A right circular cone has a height of 12 cm and a base radius of 5 cm. The cone is intersected by a plane parallel to its base, creating a smaller cone at the top with height 4 cm. What is the volume of the frustum (the remaining portion between the two parallel planes)? Answer: 700π/3 Solution: When a cone is cut by a plane parallel to its base, the resulting smaller cone is similar to the original cone. This similarity allows us to find the radius of the smaller cone using proportional relationships. The volume of a cone is calculated using the formula V = (1/3)πr²h.
   Full step-by-step solution

   When a cone is cut by a plane parallel to its base, the resulting smaller cone is similar to the original cone. This similarity allows us to find the radius of the smaller cone using proportional relationships. The volume of a cone is calculated using the formula V = (1/3)πr²h. For frustum problems, we typically calculate the volume of the original solid and subtract the volume of the removed portion.

7. 2^(x+1) = 32 Answer: 4 Solution: Write down the given equation. 2^(x+1) = 32 Recognize that both sides of the equation can be written as powers of 2. We know that 32 is the same as 2^5 because 2 * 2 * 2 * 2 * 2 = 32.
   Full step-by-step solution

   Step 1: Write down the given equation. 2^(x+1) = 32 Step 2: Recognize that both sides of the equation can be written as powers of 2. We know that 32 is the same as 2^5 because 2 * 2 * 2 * 2 * 2 = 32. So we rewrite the equation as: 2^(x+1) = 2^5 Step 3: Since the bases are the same (both are base 2), we can set the exponents equal to each other. This gives us the new equation: x + 1 = 5 Step 4: Solve for x by subtracting 1 from both sides of the equation. x + 1 - 1 = 5 - 1 x = 4 Step 5: State the final answer. The solution is x = 4.