Exponential Form

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Answer Key & Explanations

Exponential Form · Grade 9 · Worksheet 1

1. A bacterial culture starts with 500 bacteria and doubles every 3 hours. The population can be modeled by the function P(t) = 500 × 2^(t/3), where t is time in hours. How many bacteria will there be after 9 hours? Answer: 4000 Solution: Initial bacteria count = 500 Doubling time = 3 hours Population model: P(t) = 500 × 2^(t/3) t = time in hours Identify the time given in the problem. t = 9 hours Substitute t = 9 into the formula.
   Full step-by-step solution

   Let's solve this step by step. We are given: Initial bacteria count = 500 Doubling time = 3 hours Population model: P(t) = 500 × 2^(t/3) t = time in hours Step 1: Identify the time given in the problem. t = 9 hours Step 2: Substitute t = 9 into the formula. P(9) = 500 × 2^(9/3) Step 3: Simplify the exponent. 9/3 = 3 So P(9) = 500 × 2^3 Step 4: Calculate 2^3. 2^3 = 2 × 2 × 2 = 8 Step 5: Multiply 500 by 8. 500 × 8 = 4000 Step 6: Interpret the result. After 9 hours, the bacteria population will be 4000. This makes sense because every 3 hours the population doubles: At t = 0: 500 At t = 3: 1000 At t = 6: 2000 At t = 9: 4000 Final answer: 4000

2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). The triangle is then reflected across the line y = x. What is the area of the new triangle formed after reflection? Answer: 24 Solution: Identify the original triangle's vertices: A(0,0), B(6,0), C(6,8) Apply reflection across y = x by swapping x and y coordinates: A'(0,0), B'(0,6), C'(8,6) Calculate the area of the original triangle: base = 6, height = 8, area = (1/2) × 6 × 8 = 24 Reflection preserves area, so the new triangle…
   Full step-by-step solution

   Step 1: Identify the original triangle's vertices: A(0,0), B(6,0), C(6,8) Step 2: Apply reflection across y = x by swapping x and y coordinates: A'(0,0), B'(0,6), C'(8,6) Step 3: Calculate the area of the original triangle: base = 6, height = 8, area = (1/2) × 6 × 8 = 24 Step 4: Reflection preserves area, so the new triangle also has area = 24 The answer is 24.

3. Mere's investment of $400 grows by 6% each year. Write the exponential function f(t) = a·b^t that models this growth. Answer: 400·1.06^t Solution: Identify the initial amount a. The investment starts at $400, so a = 400. Determine the growth factor b.
   Full step-by-step solution

   Step 1: Identify the initial amount a. The investment starts at $400, so a = 400. Step 2: Determine the growth factor b. A 6% growth means the amount multiplies by (1 + 0.06) = 1.06 each year, so b = 1.06. Step 3: Write the function in the form f(t) = a·b^t. Step 4: Substitute the values: f(t) = 400·1.06^t Step 5: The final exponential function is f(t) = 400·1.06^t

4. 2^{x+1} = 32 Answer: 4 Solution: We are given: 2^(x+1) = 32 Recognize that 32 is a power of 2. 32 = 2 × 2 × 2 × 2 × 2 = 2^5. 2^(x+1) = 2^5.
   Full step-by-step solution

   We are given: 2^(x+1) = 32 Step 1: Recognize that 32 is a power of 2. 32 = 2 × 2 × 2 × 2 × 2 = 2^5. So we can rewrite the equation as: 2^(x+1) = 2^5. Step 2: Since the bases are the same (base 2) and are positive and not equal to 1, we can set the exponents equal to each other. Therefore: x + 1 = 5. Step 3: Solve for x. Subtract 1 from both sides: x = 5 - 1 x = 4. Step 4: Check the solution. Substitute x = 4 into the original equation: 2^(4+1) = 2^5 = 32, which matches the right-hand side. Thus, the correct answer is x = 4.

5. 2^(x+3) = 32 = ? Answer: 2 Solution: Write 32 as a power of 2: 32 = 2^5 Substitute into the equation: 2^(x+3) = 2^5 Since the bases are equal, set the exponents equal: x + 3 = 5 Solve for x: x = 5 - 3 x = 2 The answer is 2.
   Full step-by-step solution

   Step 1: Write 32 as a power of 2: 32 = 2^5 Step 2: Substitute into the equation: 2^(x+3) = 2^5 Step 3: Since the bases are equal, set the exponents equal: x + 3 = 5 Step 4: Solve for x: x = 5 - 3 Step 5: x = 2 The answer is 2.

6. Lena is studying the spread of a social media post. When she first shares it, 15 people see it immediately. The number of people who see the post triples every 2 hours. Write an exponential function in the form V(t) = a * b^t that models the number of viewers V after t hours. Answer: V(t) = 15 * (sqrt(3))^t Solution: Identify the initial value 'a'. The problem states 15 people see it immediately, so a = 15. Determine the base 'b'.
   Full step-by-step solution

   Step 1: Identify the initial value 'a'. The problem states 15 people see it immediately, so a = 15. Step 2: Determine the base 'b'. The number of viewers triples every 2 hours. This means that after 2 hours, V(2) = 15 * 3. Step 3: Using the function form V(t) = a * b^t, we can write V(2) = 15 * b^2. Step 4: Set the two expressions for V(2) equal: 15 * b^2 = 15 * 3. Step 5: Divide both sides by 15: b^2 = 3. Step 6: Solve for b: b = sqrt(3). Step 7: Write the final function: V(t) = 15 * (sqrt(3))^t.

7. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A circle is circumscribed around this triangle, passing through all three vertices. What is the area of the circumscribed circle? (Use π = 3.14) Answer: 78.5 Solution: Identify that for a right triangle, the circumscribed circle has its center at the midpoint of the hypotenuse and the hypotenuse is the diameter. Find the hypotenuse length using the distance formula between points (0,0) and (6,8).
   Full step-by-step solution

   Step 1: Identify that for a right triangle, the circumscribed circle has its center at the midpoint of the hypotenuse and the hypotenuse is the diameter. Step 2: Find the hypotenuse length using the distance formula between points (0,0) and (6,8). Step 3: Calculate: sqrt((6-0)^2 + (8-0)^2) = sqrt(36 + 64) = sqrt(100) = 10 units. Step 4: The diameter is 10 units, so the radius is 10 ÷ 2 = 5 units. Step 5: Calculate the area using A = πr² = 3.14 × 5² = 3.14 × 25 = 78.5 square units. The answer is 78.5.

8. Sophia's investment of $800 grows at 7% annually. Write f(t) = a·b^t Answer: 800·1.07^t Solution: The initial investment is $800, so a = 800 The growth rate is 7%, which means the investment multiplies by 1 + 0.07 = 1.07 each year Therefore, b = 1.07 The exponential function is f(t) = 800·1.07^t
   Full step-by-step solution

   Step 1: The initial investment is $800, so a = 800 Step 2: The growth rate is 7%, which means the investment multiplies by 1 + 0.07 = 1.07 each year Step 3: Therefore, b = 1.07 Step 4: The exponential function is f(t) = 800·1.07^t

9. Isabella's investment of $800 grows by 9% each year. Write the function in the form f(t) = a·b^t Answer: 800·1.09^t Solution: Identify the initial amount a. The investment starts at $800, so a = 800. Determine the growth factor b.
   Full step-by-step solution

   Step 1: Identify the initial amount a. The investment starts at $800, so a = 800. Step 2: Determine the growth factor b. The investment grows by 9% each year, which means it multiplies by 1 + 0.09 = 1.09 each year, so b = 1.09. Step 3: Write the function in the form f(t) = a·b^t. Substituting the values gives f(t) = 800·1.09^t. The answer is 800·1.09^t.