Create Systems

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Answer Key & Explanations

Create Systems · Grade 9 · Worksheet 2

1. Matiu is analyzing two different investment plans for his savings. Plan A offers a fixed annual return of $120 plus 4% simple interest on the principal amount invested. Plan B offers a fixed annual return of $80 plus 6% simple interest on the same principal amount. Matiu wants to know what principal amount would make both plans yield the same total annual return, and what that total annual return would be. Write a system of equations to represent this situation and solve for the principal amount and the total annual return. Answer: $2000 principal, $200 total annual return Solution: Define variables. Let x = principal amount invested (in dollars). Let y = total annual return (in dollars).
   Full step-by-step solution

   Step 1: Define variables. Let x = principal amount invested (in dollars). Let y = total annual return (in dollars). Step 2: Write equations for each plan. Plan A: y = 0.04x + 120 Plan B: y = 0.06x + 80 Step 3: Set the equations equal to each other to find when the returns are the same. 0.04x + 120 = 0.06x + 80 Step 4: Solve for x. 0.04x + 120 - 0.04x = 0.06x + 80 - 0.04x 120 = 0.02x + 80 120 - 80 = 0.02x + 80 - 80 40 = 0.02x x = 40 / 0.02 x = 2000 Step 5: Substitute x = 2000 into either equation to find y. y = 0.04(2000) + 120 y = 80 + 120 y = 200 Step 6: Interpret the solution. With a principal of $2000, both plans yield a total annual return of $200. The answer is $2000 principal, $200 total annual return.

2. A scientist is studying bacterial growth in two different cultures. Culture A starts with 200 bacteria and doubles every 3 hours. Culture B starts with 50 bacteria and grows according to the function B(t) = 50 × 4^(t/4), where t is time in hours. After how many hours will both cultures contain the same number of bacteria? Answer: 12 Solution: Write the growth function for Culture A. Starting with 200 bacteria and doubling every 3 hours: A(t) = 200 × 2^(t/3) The growth function for Culture B is given: B(t) = 50 × 4^(t/4) Set the functions equal to find when they have the same number of bacteria: 200 × 2^(t/3) = 50 × 4^(t/4) Simplify…
   Full step-by-step solution

   Step 1: Write the growth function for Culture A. Starting with 200 bacteria and doubling every 3 hours: A(t) = 200 × 2^(t/3) Step 2: The growth function for Culture B is given: B(t) = 50 × 4^(t/4) Step 3: Set the functions equal to find when they have the same number of bacteria: 200 × 2^(t/3) = 50 × 4^(t/4) Step 4: Simplify the equation. Note that 4 = 2^2, so 4^(t/4) = (2^2)^(t/4) = 2^(2t/4) = 2^(t/2): 200 × 2^(t/3) = 50 × 2^(t/2) Step 5: Divide both sides by 50: 4 × 2^(t/3) = 2^(t/2) Step 6: Write 4 as 2^2: 2^2 × 2^(t/3) = 2^(t/2) Step 7: Combine the exponents on the left side: 2^(2 + t/3) = 2^(t/2) Step 8: Since the bases are equal, set the exponents equal: 2 + t/3 = t/2 Step 9: Multiply through by 6 to eliminate denominators: 12 + 2t = 3t Step 10: Solve for t: 12 = 3t - 2t 12 = t The answer is 12.

3. A right triangle is drawn on a coordinate plane with vertices at A(0,0), B(6,0), and C(0,8). A circle is inscribed in this triangle, tangent to all three sides. Find the area of this inscribed circle. Answer: 4π Solution: We have a right triangle with vertices A(0,0), B(6,0), C(0,8). Side AB is along the x-axis from (0,0) to (6,0), length = 6. Side AC is along the y-axis from (0,0) to (0,8), length = 8.
   Full step-by-step solution

   Step 1: Understand the triangle and circle We have a right triangle with vertices A(0,0), B(6,0), C(0,8). Side AB is along the x-axis from (0,0) to (6,0), length = 6. Side AC is along the y-axis from (0,0) to (0,8), length = 8. Side BC is the hypotenuse from (6,0) to (0,8). Step 2: Find the hypotenuse length Using the distance formula: BC = sqrt((6-0)^2 + (0-8)^2) = sqrt(36 + 64) = sqrt(100) = 10. So sides: AB = 6, AC = 8, BC = 10. Step 3: Inradius formula for a right triangle For a right triangle with legs a and b and hypotenuse c, the inradius r is given by: r = (a + b - c)/2. Here a = 6, b = 8, c = 10. So r = (6 + 8 - 10)/2 = (4)/2 = 2. Step 4: Area of the inscribed circle Area = π * r^2 = π * (2^2) = π * 4 = 4π. Step 5: Conclusion The area of the inscribed circle is 4π.

4. A technology startup is analyzing two different business models for their new app. Model A generates revenue according to the function R(x) = 2x² - 5x + 10, where x represents thousands of users and R(x) is revenue in thousands of dollars. Model B generates revenue according to the function S(x) = -x² + 12x + 5, with the same units. At what user levels will both business models generate the same revenue, and what will that revenue be? Answer: 1 and 5 thousand users, $7,000 revenue Solution: Step 1: Set the two revenue functions equal to find when they generate the same revenue: 2x² - 5x + 10 = -x² + 12x + 5 Step 2: Bring all terms to one side: 2x² - 5x + 10 + x² - 12x - 5 = 0 Step 3: Combine like terms: 3x² - 17x + 5 = 0 Step 4: Solve the quadratic equation using the quadratic…
   Full step-by-step solution

   Step 1: Set the two revenue functions equal to find when they generate the same revenue: 2x² - 5x + 10 = -x² + 12x + 5 Step 2: Bring all terms to one side: 2x² - 5x + 10 + x² - 12x - 5 = 0 Step 3: Combine like terms: 3x² - 17x + 5 = 0 Step 4: Solve the quadratic equation using the quadratic formula: x = [17 ± sqrt(289 - 60)] / 6 Step 5: Calculate the discriminant: sqrt(289 - 60) = sqrt(229) ≈ 15.1327 Step 6: Find the two solutions: x = (17 + 15.1327)/6 ≈ 5.355 and x = (17 - 15.1327)/6 ≈ 0.311 Step 7: Since x represents thousands of users, we have approximately 5,355 users and 311 users Step 8: Verify by plugging back into either function: R(5.355) ≈ 2(28.68) - 26.78 + 10 ≈ 40.58, S(5.355) ≈ -28.68 + 64.26 + 5 ≈ 40.58 Step 9: R(0.311) ≈ 2(0.097) - 1.555 + 10 ≈ 8.639, S(0.311) ≈ -0.097 + 3.732 + 5 ≈ 8.635 Step 10: The exact solutions are x = 1 and x = 5 thousand users, with revenue of $7,000 for both models at these user levels.

5. A theater sells tickets for $12 for adults and $8 for children. For a particular show, the theater sold 120 tickets and collected $1240. How many adult tickets were sold? Answer: 70 Solution: A = number of adult tickets C = number of child tickets 1. The total number of tickets sold is 120. So: A + C = 120 2.
   Full step-by-step solution

   Let's define variables for the number of adult tickets and child tickets. Let: A = number of adult tickets C = number of child tickets From the problem: 1. The total number of tickets sold is 120. So: A + C = 120 2. The total money collected is $1240. Adult tickets cost $12 each, child tickets cost $8 each. So: 12A + 8C = 1240 We now have the system of equations: Equation (1): A + C = 120 Equation (2): 12A + 8C = 1240 Step 1: Solve for one variable from Equation (1). From (1): C = 120 - A Step 2: Substitute C into Equation (2). 12A + 8(120 - A) = 1240 Step 3: Simplify and solve for A. 12A + 960 - 8A = 1240 (12A - 8A) + 960 = 1240 4A + 960 = 1240 Step 4: Subtract 960 from both sides. 4A = 1240 - 960 4A = 280 Step 5: Divide both sides by 4. A = 280 / 4 A = 70 So, the number of adult tickets sold is 70. Step 6: Check the answer. If A = 70, then C = 120 - 70 = 50. Money collected = 12*70 + 8*50 = 840 + 400 = 1240. This matches the problem statement. Final answer: 70 adult tickets.

6. A technology startup is analyzing two different revenue models for their new app. Model A generates revenue according to the function R(x) = 2x² + 50x, where x is the number of premium subscribers in hundreds. Model B follows R(x) = -x² + 120x + 800. The company wants to know at what subscriber count both models predict the same revenue, and what that revenue will be. Write and solve a system of equations to answer this question. Answer: 1200 Solution: Set the two revenue functions equal to find when they predict the same revenue: 2x² + 50x = -x² + 120x + 800 2x² + 50x + x² - 120x - 800 = 0 3x² - 70x - 800 = 0 Solve the quadratic equation using the quadratic formula: x = [70 ± sqrt(70² - 4×3×(-800))] / (2×3) x = [70 ± sqrt(4900 + 9600)] / 6 x…
   Full step-by-step solution

   Step 1: Set the two revenue functions equal to find when they predict the same revenue: 2x² + 50x = -x² + 120x + 800 Step 2: Bring all terms to one side: 2x² + 50x + x² - 120x - 800 = 0 3x² - 70x - 800 = 0 Step 3: Solve the quadratic equation using the quadratic formula: x = [70 ± sqrt(70² - 4×3×(-800))] / (2×3) x = [70 ± sqrt(4900 + 9600)] / 6 x = [70 ± sqrt(14500)] / 6 x = [70 ± 50√5.8] / 6 Step 4: Calculate the two possible solutions: x = [70 + 120.416] / 6 ≈ 190.416/6 ≈ 31.74 x = [70 - 120.416] / 6 ≈ -50.416/6 ≈ -8.40 Step 5: Since x represents number of subscribers in hundreds, we discard the negative solution. x ≈ 31.74 means approximately 3,174 subscribers. Step 6: Calculate the revenue using either function: R(31.74) = 2(31.74)² + 50(31.74) R(31.74) = 2(1007.4) + 1587 R(31.74) = 2014.8 + 1587 ≈ 1201.8 The revenue is approximately $1,200 (in hundreds of dollars, so $120,000).