curiouskids

Due: Aug 6th

Questions

  1. Solve the system of equations: x^2 + y^2 = 25 x + y = 10
  2. Solve the equation for y: x^2 + y^2 = 49
  3. Solve the inequality: 2x^2 - 3x - 2 > 0
  4. If a car travels at a constant speed of 60 mph, how far will it travel in 4.5 hours?
  5. In triangle ABC, angle A is 45 degrees and angle B is 60 degrees. If the side opposite angle A is 7 units, what is the length of the side opposite angle B?
  6. If the complex number z = 4 + 3i, find the absolute value of z.
  7. If a vector v has a magnitude of 8 units and makes an angle of 40 degrees with the x-axis, what are its components?
  8. 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?
  9. Simplify the expression: (3x^2)^2 - (2x)^2
  10. 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?
  11. Evaluate the sine, cosine, and tangent of angle θ if θ is 30 degrees.
  12. Find the area of a triangle with vertices at points A(1,2), B(4,6), and C(5,1).

Solutions

  1. The system of equations x^2 + y^2 = 25 x + y = 10 can be solved using substitution or numerical methods. A possible solution is x = 5, y = 5.

  2. To solve x^2 + y^2 = 49 for y, you rearrange it to get y = sqrt(49 - x^2).

  3. The inequality 2x^2 - 3x - 2 > 0 factors to (2x + 1)(x - 2) > 0. The solution is x < -1/2 or x > 2.

  4. If a car travels at a constant speed of 60 mph, it will travel 60 * 4.5 = 270 miles in 4.5 hours.

  5. In triangle ABC, angle A is 45 degrees and angle B is 60 degrees. By the Law of Sines, the length of the side opposite angle B is (7/sin(45)) * sin(60) = 7 * sqrt(2) units.

  6. The absolute value of a complex number z = 4 + 3i is sqrt((4)^2 + (3)^2) = 5.

  7. The components of a vector v with a magnitude of 8 units and making an angle of 40 degrees with the x-axis are v_x = 8 * cos(40) = 6.13 and v_y = 8 * sin(40) = 5.14.

  8. 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.

  9. The simplified expression of (3x^2)^2 - (2x)^2 is 9x^4 - 4x^2.

  10. 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.

  11. If angle θ is 30 degrees, then sin(θ) = 1/2, cos(θ) = sqrt(3)/2, and tan(θ) = sqrt(3)/3.

  12. The area of a triangle with vertices at points A(1,2), B(4,6), and C(5,1) can be calculated by using the formula for the area of a triangle given its vertices, A = 1/2 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)| = 4.5 square units.

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