Equivalent Forms

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Equivalent Forms · Grade 9 · Worksheet 1

1. A research scientist is studying the decay of a radioactive isotope. The amount remaining after t years is modeled by the exponential function A(t) = 800 × (1/2)^(t/12). The scientist needs to determine when only 100 grams of the isotope will remain. How many years will it take for the isotope to decay to this amount? Answer: 36 Solution: Step 1: Set up the equation using the given function: 800 × (1/2)^(t/12) = 100 Step 2: Divide both sides by 800: (1/2)^(t/12) = 100/800 = 1/8 Step 3: Recognize that 1/8 = (1/2)^3 Step 4: Set the exponents equal: t/12 = 3 Step 5: Multiply both sides by 12: t = 3 × 12 = 36 Step 6: Verify by…
   Full step-by-step solution

   Step 1: Set up the equation using the given function: 800 × (1/2)^(t/12) = 100 Step 2: Divide both sides by 800: (1/2)^(t/12) = 100/800 = 1/8 Step 3: Recognize that 1/8 = (1/2)^3 Step 4: Set the exponents equal: t/12 = 3 Step 5: Multiply both sides by 12: t = 3 × 12 = 36 Step 6: Verify by substituting back: 800 × (1/2)^(36/12) = 800 × (1/2)^3 = 800 × 1/8 = 100 The answer is 36 years.

2. A construction company is designing a suspension bridge where the cable shape follows a parabolic curve. The cable's height above the roadway is modeled by the function h(x) = -0.02x² + 0.8x + 10, where x is the horizontal distance in meters from the left tower and h(x) is the height in meters. The engineers need to find the horizontal distance from the left tower where the cable reaches its maximum height. At what distance x does the cable reach its maximum height? Answer: 20 Solution: The cable height is given by h(x) = -0.02x² + 0.8x + 10 This is a quadratic function in standard form ax² + bx + c, where a = -0.02, b = 0.8, and c = 10 Since a is negative (-0.02), the parabola opens downward, so the vertex represents the maximum point The x-coordinate of the vertex is given by…
   Full step-by-step solution

   Step 1: The cable height is given by h(x) = -0.02x² + 0.8x + 10 Step 2: This is a quadratic function in standard form ax² + bx + c, where a = -0.02, b = 0.8, and c = 10 Step 3: Since a is negative (-0.02), the parabola opens downward, so the vertex represents the maximum point Step 4: The x-coordinate of the vertex is given by x = -b/(2a) Step 5: Substitute the values: x = -0.8/(2 × -0.02) = -0.8/(-0.04) = 20 Step 6: Therefore, the cable reaches its maximum height at x = 20 meters from the left tower The answer is 20.

3. A tech company is modeling the depreciation of their computer equipment using the function V(t) = 1200 × (0.85)^t, where V(t) represents the value in dollars after t years. Their accountant needs to rewrite this function in the form V(t) = 1200 × e^(kt) to match their financial software. Which expression is equivalent to the original depreciation function? Answer: B. 1200 × e^(-0.1625t) Solution: Start with the original function: V(t) = 1200 × (0.85)^t We need to convert (0.85)^t to the form e^(kt) Using the property a^t = e^(t × ln(a)), we get (0.85)^t = e^(t × ln(0.85)) Calculate ln(0.85) using a calculator: ln(0.85) ≈ -0.1625 Therefore, (0.85)^t = e^(-0.1625t) Substitute back into the…
   Full step-by-step solution

   Step 1: Start with the original function: V(t) = 1200 × (0.85)^t Step 2: We need to convert (0.85)^t to the form e^(kt) Step 3: Using the property a^t = e^(t × ln(a)), we get (0.85)^t = e^(t × ln(0.85)) Step 4: Calculate ln(0.85) using a calculator: ln(0.85) ≈ -0.1625 Step 5: Therefore, (0.85)^t = e^(-0.1625t) Step 6: Substitute back into the original function: V(t) = 1200 × e^(-0.1625t) Step 7: Compare with the answer choices - this matches choice C The correct answer is 1200 × e^(-0.1625t).

4. A rectangular garden has a length that is 5 meters more than twice its width. If the area of the garden is 63 square meters, what is the width of the garden in meters? Answer: 4.5 Solution: Let’s go step-by-step. Let the width of the garden be \( w \) meters. The length is 5 meters more than twice the width, so: length \( l = 2w + 5 \).
   Full step-by-step solution

   Let’s go step-by-step. --- **Step 1: Define the variables** Let the width of the garden be \( w \) meters. The length is 5 meters more than twice the width, so: length \( l = 2w + 5 \). --- **Step 2: Write the area equation** Area of rectangle = length × width. Given area = 63 m², so: \[ l \times w = 63 \] \[ (2w + 5) \times w = 63 \] --- **Step 3: Expand and rearrange** \[ 2w^2 + 5w = 63 \] \[ 2w^2 + 5w - 63 = 0 \] --- **Step 4: Solve the quadratic equation** Use the quadratic formula: \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here \( a = 2 \), \( b = 5 \), \( c = -63 \). First, discriminant: \[ b^2 - 4ac = 25 - 4(2)(-63) = 25 + 504 = 529 \] \[ \sqrt{529} = 23 \] So: \[ w = \frac{-5 \pm 23}{2 \times 2} = \frac{-5 \pm 23}{4} \] --- **Step 5: Two possible solutions** First: \( w = \frac{-5 + 23}{4} = \frac{18}{4} = 4.5 \) Second: \( w = \frac{-5 - 23}{4} = \frac{-28}{4} = -7 \) Width cannot be negative, so \( w = 4.5 \). --- **Step 6: Check** Width \( w = 4.5 \) m Length \( l = 2(4.5) + 5 = 9 + 5 = 14 \) m Area = \( 14 \times 4.5 = 63 \) m² ✔ --- **Final answer:** The width is 4.5 meters.

5. 2^(3x-1) = 32 Answer: 2 Solution: Recognize that 32 can be written as a power of 2: 32 = 2^5 Rewrite the equation: 2^(3x-1) = 2^5 Since the bases are equal, set the exponents equal: 3x - 1 = 5 Add 1 to both sides: 3x = 6 Divide both sides by 3: x = 2 The answer is 2.
   Full step-by-step solution

   Step 1: Recognize that 32 can be written as a power of 2: 32 = 2^5 Step 2: Rewrite the equation: 2^(3x-1) = 2^5 Step 3: Since the bases are equal, set the exponents equal: 3x - 1 = 5 Step 4: Add 1 to both sides: 3x = 6 Step 5: Divide both sides by 3: x = 2 The answer is 2.

6. Which form of the quadratic expression 4x² + 20x + 25 reveals its minimum value, and what is that minimum value? Answer: 0 Solution: Recognize that 4x² + 20x + 25 can be written as (2x)² + 2(2x)(5) + 5², which is a perfect square trinomial. Factor it as (2x + 5)².
   Full step-by-step solution

   Step 1: Recognize that 4x² + 20x + 25 can be written as (2x)² + 2(2x)(5) + 5², which is a perfect square trinomial. Step 2: Factor it as (2x + 5)². Step 3: The expression (2x + 5)² is always greater than or equal to 0, with its minimum value occurring when 2x + 5 = 0. Step 4: Solve 2x + 5 = 0 → 2x = -5 → x = -5/2. Step 5: Substitute x = -5/2 into (2x + 5)²: (2(-5/2) + 5)² = (-5 + 5)² = 0² = 0. The minimum value is 0.

7. Which form of the quadratic function reveals the y-intercept: y = 2(x - 3)^2 + 8 or y = 2x^2 - 12x + 26? Answer: y = 2x^2 - 12x + 26 Solution: The y-intercept is the value of y when x = 0. Step 2: For the vertex form y = 2(x - 3)^2 + 8, substitute x = 0: y = 2(0 - 3)^2 + 8 = 2(9) + 8 = 18 + 8 = 26. This requires calculation.
   Full step-by-step solution

   Step 1: The y-intercept is the value of y when x = 0. Step 2: For the vertex form y = 2(x - 3)^2 + 8, substitute x = 0: y = 2(0 - 3)^2 + 8 = 2(9) + 8 = 18 + 8 = 26. This requires calculation. Step 3: For the standard form y = 2x^2 - 12x + 26, substitute x = 0: y = 2(0)^2 - 12(0) + 26 = 0 - 0 + 26 = 26. The constant term directly gives the y-intercept. Step 4: Therefore, the standard form y = 2x^2 - 12x + 26 reveals the y-intercept (0, 26) immediately. The answer is y = 2x^2 - 12x + 26.