curiouskids

Due: Aug 2nd

Questions

Also a tad harder

  1. Solve the system of equations: 2x^2 + y^2 = 25 2x + y = 9
  2. Solve the equation for y: 4x^2 + y^2 = 16
  3. Solve the inequality: 2x^2 + 3 > 3(x^2 + 1)
  4. A car uses 10 liters of fuel for every 100 kilometers it travels. The fuel efficiency of the car decreases by 1.5% for every additional 100kg of weight. How much fuel will the car use to travel 800 kilometers if its weight increases by 400kg?
  5. A rectangular garden has a length that is 4m less than twice its width. The area of the garden is 72m^2. What are the dimensions of the garden?
  6. You have a 20% salt solution and a 60% salt solution. How much of each should be mixed to create 400ml of a 50% salt solution?
  7. A car and a truck set off from the same place, heading in the same direction. The car travels at a speed that is 20km/h faster than the truck. After 2 hours, the car is 20km ahead of the truck. What is the speed of the car and truck?
  8. A plane climbs at an angle of 30 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: 2x^3 + 3x^2 - 81x = 0
  10. The sum of the interior angles of a regular polygon is five times the sum of its exterior angles. How many sides does the polygon have?
  11. Simplify the expression: 3x^3 + 2x^2 - x + 5 - (2x^3 - x^2 + 2x - 3)
  12. You invested $30,000 between two accounts. The first account pays 4% annual interest compounded quarterly and the second pays 5% annual interest compounded annually. If you earned $1,350 in interest after one year, how much did you invest in each account?

Solutions

  1. The system of equations 2x^2 + y^2 = 25 2x + y = 9 can be solved using substitution or numerical methods. A possible solution is x = 3, y = 3.

  2. To solve 4x^2 + y^2 = 16 for y, you rearrange it to get y = sqrt(16 - 4x^2).

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

  4. If the car’s weight increases by 400 kg, the fuel efficiency decreases by 1.5% * 4 = 6%. Therefore, the new fuel efficiency is 10 * 1.06 = 10.6 liters/100km. To travel 800 kilometers, the car would use 10.6 * 8 = 84.8 liters.

  5. Let’s denote the width as x and the length as 2x - 4. The area of the garden is x * (2x - 4) = 72. Solving for x gives x = 6 and 2x - 4 = 8. Therefore, the width is 6m and the length is 8m.

  6. Let’s denote the amount of the 20% solution as x and the amount of the 60% solution as 400 - x. The equation is 0.2x + 0.6(400 - x) = 0.5 * 400. Solving for x gives x = 200 ml. Therefore, 200 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 truck as x - 20. We have two equations: 2x = 2(x - 20) + 20 and x > 0, x - 20 > 0. Solving this system gives x = 40 km/h for the car and x - 20 = 20 km/h for the truck.

  8. Using the trigonometric relation tan(θ) = opposite / adjacent, the height of the plane above the ground is 300 * tan(30) = 173 km.

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

  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) = 5 * 360, then n = 12. The polygon has 12 sides.

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