curiouskids

Due: Jul 30th

Questions

  1. Solve the system of equations: 2x^2 - 3y = 20 x + 4y^2 = 16
  2. Solve the equation for y: 5x^2 - 3y^2 = 27
  3. Solve the inequality: 4x^2 - 6 > 3(x^2 - 2) + 2
  4. A car’s fuel efficiency decreases by 2% for every 100kg increase in weight. If the car initially has a fuel efficiency of 15 km/l for a weight of 1200kg, what would be its fuel efficiency if the weight becomes 1500kg?
  5. A rectangular garden has a length that is 2m more than three times its width. If the area of the garden is 100m^2, find the dimensions of the garden.
  6. You have a 40% alcohol solution and a 70% alcohol solution. How much of each should be mixed to create 500ml of a 60% alcohol solution?
  7. If a cyclist travels at an average speed that is 1km/h faster than a runner and both cover a distance of 20km, but the cyclist takes one hour less than the runner, what is the average speed of the cyclist and runner?
  8. A plane takes off at an angle of 15 degrees and then flies straight for 200 kilometers. How high is the plane above the ground, to the nearest kilometer?
  9. Solve for x: 2x^3 - 3x^2 - 36x = 0
  10. The sum of the interior angles of a regular polygon is four times the sum of its exterior angles. How many sides does the polygon have?
  11. Simplify the expression: 2x^3 - 3x^2 + 4x - 5 - (x^3 - 2x^2 + 3x - 4)
  12. You invested $20,000 between two accounts. The first account pays 4.5% annual interest compounded semi-annually and the second pays 5% annual interest compounded annually. If you earned $980 in interest after one year, how much did you invest in each account?

Solutions

  1. This system of equations is non-linear and may be solved using substitution or numerical methods. A possible solution is x = 4, y = 2.

  2. To solve 5x^2 - 3y^2 = 27 for y, you rearrange it to get y = sqrt((5x^2 - 27)/3).

  3. Solving the inequality 4x^2 - 6 > 3(x^2 - 2) + 2 gives x > sqrt(10/3) or x < -sqrt(10/3).

  4. If the car’s weight increases by 300 kg, the fuel efficiency decreases by 2% * 3 = 6%. Therefore, the new fuel efficiency would be 15 * (1 - 0.06) = 14.1 km/l.

  5. Let’s denote the width as x and the length as 3x + 2. The area of the garden is x * (3x + 2) = 100. Solving for x gives x = 5 and 3x + 2 = 17. Therefore, the width is 5m and the length is 17m.

  6. Let’s denote the amount of the 40% solution as x and the amount of the 70% solution as 500 - x. The equation is 0.4x + 0.7(500 - x) = 0.6 * 500. Solving for x gives x = 250 ml. Therefore, 250 ml of each solution should be mixed.

  7. Let’s denote the average speed of the cyclist as x and the average speed of the runner as x - 1. We have two equations: 20/x = 20/(x - 1) + 1 and x > 0, x - 1 > 0. Solving this system gives x = 10 km/h for the cyclist and x - 1 = 9 km/h for the runner.

  8. Using the trigonometric relation tan(θ) = opposite / adjacent, the height of the plane above the ground is 200 * tan(15) = 52 km.

  9. To solve for x in 2x^3 - 3x^2 - 36x = 0, factor to get x(2x - 9)(x + 4) = 0. This gives x = 0, x = 9/2, 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) = 4 * 360, then n = 10. The polygon has 10 sides.

  11. Simplifying the expression 2x^3 - 3x^2 + 4x - 5 - (x^3 - 2x^2 + 3x - 4) gives x^3 - x^2 + x - 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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