curiouskids

Due: Aug 4th

Questions

Tricky!

  1. Solve the system of equations: 3x^2 - 2y^2 = 20 x + 3y = 7
  2. Solve the equation for y: 3x^2 - y^2 = 12
  3. Solve the inequality: x^2 - 4 > 2(x - 2)
  4. A car uses 8 liters of fuel for every 100 kilometers it travels. The fuel efficiency of the car decreases by 2% for every additional 100kg of weight. How much fuel will the car use to travel 900 kilometers if its weight increases by 300kg?
  5. A rectangular garden has a length that is 5m more than twice its width. The area of the garden is 63m^2. What are the dimensions of the garden?
  6. You have a 15% salt solution and a 55% salt solution. How much of each should be mixed to create 350ml of a 45% salt solution?
  7. A car and a bike set off from the same place, heading in the same direction. The car travels at a speed that is 15km/h faster than the bike. After 4 hours, the car is 20km ahead of the bike. What is the speed of the car and bike?
  8. A plane climbs at an angle of 25 degrees to the ground. If it flies straight for 350 kilometers, how high is it above the ground, to the nearest kilometer?
  9. Solve for x: 3x^3 - x^2 - 45x = 0
  10. The sum of the interior angles of a regular polygon is six times the sum of its exterior angles. How many sides does the polygon have?
  11. Simplify the expression: 2x^3 - x^2 + 2x - 3 - (x^3 - 2x^2 + x - 1)
  12. You invested $40,000 between two accounts. The first account pays 3% annual interest compounded quarterly and the second pays 4.5% annual interest compounded annually. If you earned $1,600 in interest after one year, how much did you invest in each account?

Solutions

  1. The system of equations 3x^2 - 2y^2 = 20 x + 3y = 7 can be solved using substitution or numerical methods. A possible solution is x = 1, y = 2.

  2. To solve 3x^2 - y^2 = 12 for y, you rearrange it to get y = sqrt(3x^2 - 12).

  3. Solving the inequality x^2 - 4 > 2(x - 2) gives x < 0 or x > 4.

  4. If the car’s weight increases by 300 kg, the fuel efficiency decreases by 2% * 3 = 6%. Therefore, the new fuel efficiency is 8 * 1.06 = 8.48 liters/100km. To travel 900 kilometers, the car would use 8.48 * 9 = 76.32 liters.

  5. Let’s denote the width as x and the length as 2x + 5. The area of the garden is x * (2x + 5) = 63. Solving for x gives x = 7 and 2x + 5 = 19. Therefore, the width is 7m and the length is 19m.

  6. Let’s denote the amount of the 15% solution as x and the amount of the 55% solution as 350 - x. The equation is 0.15x + 0.55(350 - x) = 0.45 * 350. Solving for x gives x = 150 ml. Therefore, 150 ml of the 15% solution and 200 ml of the 55% solution should be mixed.

  7. Let’s denote the average speed of the car as x and the average speed of the bike as x - 15. We have two equations: 4x = 4(x - 15) + 20 and x > 0, x - 15 > 0. Solving this system gives x = 30 km/h for the car and x - 15 = 15 km/h for the bike.

  8. Using the trigonometric relation tan(θ) = opposite / adjacent, the height of the plane above the ground is 350 * tan(25) = 161 km.

  9. To solve for x in 3x^3 - x^2 - 45x = 0, factor to get x(3x + 9)(x - 5) = 0. This gives x = 0, x = -3, and x = 5.

  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) = 6 * 360, then n = 13. The polygon has 13 sides.

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

  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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