curiouskids

Due: Aug 1st

Questions

These are a tad harder.

  1. Solve the system of equations: 3x^2 + 4y^2 = 39 x - 2y = 3
  2. Solve the equation for y: 6x^2 - 2y^2 = 30
  3. Solve the inequality: 5x^2 - 7 > 4(x^2 - 3) + 1
  4. A bus uses 15 liters of fuel for every 100 kilometers it travels. The fuel efficiency of the bus decreases by 1% for every additional 100kg of weight. How much fuel will the bus use to travel 1000 kilometers if its weight increases by 500kg?
  5. A rectangular garden has a length that is 3m more than twice its width. The area of the garden is 98m^2. What are the dimensions of the garden?
  6. You have a 30% salt solution and a 50% salt solution. How much of each should be mixed to create 300ml of a 40% salt solution?
  7. A car and a bus set off from the same place, heading in the same direction. The car travels at a speed that is 10km/h faster than the bus. After 3 hours, the car is 15km ahead of the bus. What is the speed of the car and bus?
  8. A plane climbs at an angle of 20 degrees to the ground. If it flies straight for 300 kilometers, how high is it above the ground, to the nearest kilometer?
  9. Solve for x: 3x^3 + 2x^2 - 81x = 0
  10. The sum of the interior angles of a regular polygon is seven times the sum of its exterior angles. How many sides does the polygon have?
  11. Simplify the expression: 3x^3 - 2x^2 + x - 6 - (2x^3 - x^2 + 4x - 3)
  12. You invested $25,000 between two accounts. The first account pays 5% annual interest compounded quarterly and the second pays 6% annual interest compounded annually. If you earned $1,400 in interest after one year, how much did you invest in each account?

Solutions

  1. The system of equations 3x^2 + 4y^2 = 39 x - 2y = 3 can be solved using substitution or numerical methods. A possible solution is x = 3, y = 0.

  2. To solve 6x^2 - 2y^2 = 30 for y, you rearrange it to get y = sqrt((6x^2 - 30)/(-2)).

  3. Solving the inequality 5x^2 - 7 > 4(x^2 - 3) + 1 gives x > sqrt(8/3) or x < -sqrt(8/3).

  4. If the bus’ weight increases by 500 kg, the fuel efficiency decreases by 1% * 5 = 5%. Therefore, the new fuel efficiency is 15 * 1.05 = 15.75 liters/100km. To travel 1000 kilometers, the bus would use 15.75 * 10 = 157.5 liters.

  5. Let’s denote the width as x and the length as 2x + 3. The area of the garden is x * (2x + 3) = 98. Solving for x gives x = 7 and 2x + 3 = 17. Therefore, the width is 7m and the length is 17m.

  6. Let’s denote the amount of the 30% solution as x and the amount of the 50% solution as 300 - x. The equation is 0.3x + 0.5(300 - x) = 0.4 * 300. Solving for x gives x = 150 ml. Therefore, 150 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 bus as x - 10. We have two equations: 3x = 3(x - 10) + 15 and x > 0, x - 10 > 0. Solving this system gives x = 35 km/h for the car and x - 10 = 25 km/h for the bus.

  8. Using the trigonometric relation tan(θ) = opposite / adjacent, the height of the plane above the ground is 300 * tan(20) = 112 km.

  9. To solve for x in 3x^3 + 2x^2 - 81x = 0, factor to get x(3x - 9)(x + 3) = 0. This gives x = 0, x = 3, and x = -3.

  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) = 7 * 360, then n = 14. The polygon has 14 sides.

  11. Simplifying the expression 3x^3 - 2x^2 + x - 6 - (2x^3 - x^2 + 4x - 3) gives x^3 - x^2 - 3x - 3.

  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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