Rate of Change

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Rate of Change · Grade 8 · Worksheet 1

1. A rocket's altitude during launch is modeled by the equation h(t) = 3.2 × 10^4 + 8.5 × 10^3t, where h is altitude in meters and t is time in seconds after launch. What was the rocket's initial altitude at the moment of launch? Answer: 32000 Solution: The equation is h(t) = 3.2 × 10^4 + 8.5 × 10^3t Initial altitude occurs at t = 0 seconds (the moment of launch) Substitute t = 0 into the equation: h(0) = 3.2 × 10^4 + 8.5 × 10^3(0) Simplify: h(0) = 3.2 × 10^4 + 0 Calculate 3.2 × 10^4 = 3.2 × 10,000 = 32,000 The initial altitude is 32,000 meters…
   Full step-by-step solution

   Step 1: The equation is h(t) = 3.2 × 10^4 + 8.5 × 10^3t Step 2: Initial altitude occurs at t = 0 seconds (the moment of launch) Step 3: Substitute t = 0 into the equation: h(0) = 3.2 × 10^4 + 8.5 × 10^3(0) Step 4: Simplify: h(0) = 3.2 × 10^4 + 0 Step 5: Calculate 3.2 × 10^4 = 3.2 × 10,000 = 32,000 Step 6: The initial altitude is 32,000 meters The answer is 32000.

2. Max is reading a book for a school project. The book has 338 pages, and Max reads 15 pages each day. If Max reads at a constant rate, what is the rate of change (pages per day) and the initial value (pages read before starting, which is 0)? Answer: 15 Solution: The rate of change is the number of pages read each day: 15 pages per day. The initial value is the number of pages read at the start: 0 pages.
   Full step-by-step solution

   The rate of change is the number of pages read each day: 15 pages per day. The initial value is the number of pages read at the start: 0 pages. So the rate of change is 15 and the initial value is 0.

3. A scientist is studying bacterial growth in a lab. The initial population is 200 bacteria, and the population doubles every 3 hours. Write an exponential function in the form y = a(b)^x that models the bacterial population, where x is the number of hours and y is the total population. Answer: y = 200(2)^(x/3) Solution: The problem says: initial population is 200 bacteria. In the exponential model \( y = a(b)^x \), \( a \) is the initial value when \( x = 0 \). So \( a = 200 \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the initial value** The problem says: initial population is 200 bacteria. In the exponential model \( y = a(b)^x \), \( a \) is the initial value when \( x = 0 \). So \( a = 200 \). --- **Step 2: Determine the growth factor** The population doubles every 3 hours. That means if we look at a time period of 3 hours, the population multiplies by 2. So the growth factor per 3 hours is \( b = 2 \). --- **Step 3: Adjust for the time unit** The formula \( y = a(b)^x \) uses \( x \) in hours, but the doubling period is 3 hours. We can't just write \( y = 200(2)^x \) because that would mean doubling every 1 hour, which is wrong. Instead, think: In \( x \) hours, the number of doubling periods is \( x/3 \). So population = initial × (growth factor per period)^(number of periods) = \( 200 \times 2^{(x/3)} \). --- **Step 4: Write the final function** We have \( a = 200 \), \( b^x \) becomes \( 2^{(x/3)} \). So \( y = 200(2)^{x/3} \). --- **Step 5: Check** If \( x = 0 \): \( y = 200(2)^{0} = 200 \) ✅ If \( x = 3 \): \( y = 200(2)^{3/3} = 200(2)^1 = 400 \) ✅ (doubled) If \( x = 6 \): \( y = 200(2)^{6/3} = 200(2)^2 = 800 \) ✅ (doubled again) --- **Final answer:** y = 200(2)^(x/3)

4. (3.6 × 10⁵) ÷ (9 × 10²) = ? Answer: 400 Solution: Divide the coefficients: 3.6 ÷ 9 = 0.4 Subtract the exponents: 5 - 2 = 3 Combine the results: 0.4 × 10³ Convert to standard form: 0.4 × 1000 = 400 The answer is 400.
   Full step-by-step solution

   Step 1: Divide the coefficients: 3.6 ÷ 9 = 0.4 Step 2: Subtract the exponents: 5 - 2 = 3 Step 3: Combine the results: 0.4 × 10³ Step 4: Convert to standard form: 0.4 × 1000 = 400 The answer is 400.

5. (5.2 × 10⁴) × (3.0 × 10⁻²) = ? Answer: 1560 Solution: Multiply the coefficients: 5.2 × 3.0 = 15.6 Add the exponents: 4 + (-2) = 2 Combine the results: 15.6 × 10² Convert to standard form: 15.6 × 100 = 1560 The answer is 1560.
   Full step-by-step solution

   Step 1: Multiply the coefficients: 5.2 × 3.0 = 15.6 Step 2: Add the exponents: 4 + (-2) = 2 Step 3: Combine the results: 15.6 × 10² Step 4: Convert to standard form: 15.6 × 100 = 1560 The answer is 1560.

6. Maya is tracking the growth of her bamboo plant. She notices it grows at a constant rate each week. After 4 weeks, the bamboo was 28 inches tall, and after 9 weeks, it was 53 inches tall. What was the initial height of the bamboo plant when Maya started tracking it? Answer: 8 Solution: Find the growth rate per week. The plant grew from 28 inches to 53 inches over a period of 5 weeks (from week 4 to week 9). Calculate the weekly growth: (53 - 28) / (9 - 4) = 25 / 5 = 5 inches per week.
   Full step-by-step solution

   Step 1: Find the growth rate per week. The plant grew from 28 inches to 53 inches over a period of 5 weeks (from week 4 to week 9). Step 2: Calculate the weekly growth: (53 - 28) / (9 - 4) = 25 / 5 = 5 inches per week. Step 3: Work backwards from the 4-week measurement to find the initial height. At week 4, the height was 28 inches. This means the plant grew for 4 weeks from the start. Step 4: Calculate the total growth over those 4 weeks: 4 weeks × 5 inches/week = 20 inches. Step 5: Subtract this growth from the week 4 height to find the initial height: 28 - 20 = 8 inches. The initial height was 8 inches.

7. (5 × 10⁸) × (3 × 10⁻⁴) = ? Answer: 150000 Solution: Multiply the coefficients: 5 × 3 = 15 Add the exponents: 8 + (-4) = 4 Combine the results: 15 × 10⁴ Convert to standard form: 15 × 10,000 = 150,000 The answer is 150000.
   Full step-by-step solution

   Step 1: Multiply the coefficients: 5 × 3 = 15 Step 2: Add the exponents: 8 + (-4) = 4 Step 3: Combine the results: 15 × 10⁴ Step 4: Convert to standard form: 15 × 10,000 = 150,000 The answer is 150000.

8. A rectangular prism is drawn with dimensions: length = 8 cm, width = 5 cm, and height = 3 cm. If you double all three dimensions to create a larger similar prism, what is the ratio of the surface area of the larger prism to the surface area of the original prism? Answer: 4 Solution: Original prism dimensions: length = 8 cm, width = 5 cm, height = 3 cm Surface area of original prism = 2(lw + lh + wh) = 2(8×5 + 8×3 + 5×3) = 2(40 + 24 + 15) = 2(79) = 158 cm² New prism dimensions after doubling: length = 16 cm, width = 10 cm, height = 6 cm Surface area of new prism = 2(16×10 +…
   Full step-by-step solution

   Step 1: Original prism dimensions: length = 8 cm, width = 5 cm, height = 3 cm Step 2: Surface area of original prism = 2(lw + lh + wh) = 2(8×5 + 8×3 + 5×3) = 2(40 + 24 + 15) = 2(79) = 158 cm² Step 3: New prism dimensions after doubling: length = 16 cm, width = 10 cm, height = 6 cm Step 4: Surface area of new prism = 2(16×10 + 16×6 + 10×6) = 2(160 + 96 + 60) = 2(316) = 632 cm² Step 5: Ratio of surface areas = 632 ÷ 158 = 4 Step 6: The ratio of the surface area of the larger prism to the original prism is 4:1, which simplifies to 4.