curiouskids

Due: Aug 5th

Questions

  1. Solve the system of equations: 4x^2 + 3y^2 = 28 2x - y = 6
  2. Solve the equation for y: 2x^2 + y^2 = 8
  3. Solve the inequality: 3x^2 - 2 > x^2 + 1
  4. A truck uses 20 liters of fuel for every 100 kilometers it travels. The fuel efficiency of the truck decreases by 3% for every additional 100kg of weight. How much fuel will the truck use to travel 600 kilometers if its weight increases by 200kg?
  5. A rectangular garden has a length that is 2m less than three times its width. The area of the garden is 50m^2. What are the dimensions of the garden?
  6. You have a 10% salt solution and a 60% salt solution. How much of each should be mixed to create 200ml of a 30% salt solution?
  7. A car and a motorbike set off from the same place, heading in the same direction. The car travels at a speed that is 10km/h faster than the motorbike. After 2 hours, the car is 10km ahead of the motorbike. What is the speed of the car and motorbike?
  8. A plane climbs at an angle of 20 degrees to the ground. If it flies straight for 150 kilometers, how high is it above the ground, to the nearest kilometer?
  9. Solve for x: x^3 - 2x^2 - 16x = 0
  10. The sum of the interior angles of a regular polygon is three times the sum of its exterior angles. How many sides does the polygon have?
  11. Simplify the expression: 3x^3 - 2x^2 + 4x - 3 - (2x^3 - x^2 + 2x - 2)
  12. You invested $50,000 between two accounts. The first account pays 4.5% annual interest compounded quarterly and the second pays 6% annual interest compounded annually. If you earned $2,500 in interest after one year, how much did you invest in each account?

Solutions

  1. The system of equations 4x^2 + 3y^2 = 28 2x - y = 6 can be solved using substitution or numerical methods. A possible solution is x = 2, y = -2.

  2. To solve 2x^2 + y^2 = 8 for y, you rearrange it to get y = sqrt(8 - 2x^2).

  3. Solving the inequality 3x^2 - 2 > x^2 + 1 gives x < -sqrt(3) or x > sqrt(3).

  4. If the truck’s weight increases by 200 kg, the fuel efficiency decreases by 3% * 2 = 6%. Therefore, the new fuel efficiency is 20 * 1.06 = 21.2 liters/100km. To travel 600 kilometers, the truck would use 21.2 * 6 = 127.2 liters.

  5. Let’s denote the width as x and the length as 3x - 2. The area of the garden is x * (3x - 2) = 50. Solving for x gives x = 5 and 3x - 2 = 13. Therefore, the width is 5m and the length is 13m.

  6. Let’s denote the amount of the 10% solution as x and the amount of the 60% solution as 200 - x. The equation is 0.1x + 0.6(200 - x) = 0.3 * 200. Solving for x gives x = 100 ml. Therefore, 100 ml of each solution should be mixed.

  7. Let’s denote the average speed of the car as x and the average speed of the motorbike as x - 10. We have two equations: 2x = 2(x - 10) + 10 and x > 0, x - 10 > 0. Solving this system gives x = 30 km/h for the car and x - 10 = 20 km/h for the motorbike.

  8. Using the trigonometric relation tan(θ) = opposite / adjacent, the height of the plane above the ground is 150 * tan(20) = 55 km.

  9. To solve for x in x^3 - 2x^2 - 16x = 0, factor to get x(x - 4)(x + 4) = 0. This gives x = 0, x = 4, and x = -4.

  10. The sum of the interior angles of a regular polygon is 180(n - 2) degrees and the sum of the exterior angles is always 360 degrees. If 180(n - 2) = 3 * 360, then n = 9. The polygon has 9 sides.

  11. Simplifying the expression 3x^3 - 2x^2 + 4x - 3 - (2x^3 - x^2 + 2x - 2) gives x^3 - x^2 + 2x - 1.

  12. This question involves solving for two variables with compound interest. It might be easier to use a financial calculator or a software tool that can handle compound interest calculations to solve this problem.

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