Can you please check my answers? Thanxs!

Write an equation that expresses the relationship. Use k as the constant of variation.

20. f varies jointly as b and the square of c.
-I got: f=kbc^2

22. r varies jointly as the square of s and the square of t.
-I got: r=ks^2t^2

Find the variation equation for the variation statement.
31. z varies jointly as y and the cube of x; z=96 when x=2 and y=6 (I don't understand this one. My answer choices are: a.y=2x^3y,b.y=2xy^3, c.y=-2x^3y, or d. y=-2xy^3)

Determine the constant of variation for the stated condition.
40. z varies jointly as x and y, and z=84 when x=28 and y=12.
I got: z=xy, 84=k*22*12, 84=336k, I divided both sides by 336 and I got k=84/336, but it can be reduced, so k=1/4

Solve the problem.

46. h varies jointly as f and g. Find h when f=27, g=12, and k=3.
I got: h=kfg, h=3*27*12, h=972

Write an equation that expresses the relationship. Use k for the constant of proportinality.

50. p varies directly as r and inversely as s.
-I got: p=kr/s

53. r varies directly as s and inversely as the square of t.
-I got: r=ks/t^2

Determine the constant of variation for the stated condition.

58. z varies directly as x and inversely as y, and z=2 when x=30 and y=60.
I got: z=kx/y, 2=k*30/60, I multiplied each side by 60, and got 120=30k, I divided each side by 30, and got k=120/30, it reduces to k=4

60. z varies directly as x and inversely as y, and z=4 when x=77 and y=28.
-I got:z=kx/y, 4=k*77/25, I multiplied each side by 25 and got 112=77k, I divided both sides by 77, and got k=16/11.

Solve the problem.

66. y varies jointly as a and b inversely as the square root of c, y=24 when a=6,b=4, and c=49. Find y when a=2, b=5, and c=16.
-I got: y=kab/ sqrt of c, y=2*5/ sqrt of 16, y=10/4, which reduces to y=2.5

Let's go through the answers one by one and check them:

20. f varies jointly as b and the square of c. - Your answer of f = kbc^2 is correct.

22. r varies jointly as the square of s and the square of t. - Your answer of r = ks^2t^2 is correct.

31. z varies jointly as y and the cube of x; z = 96 when x = 2 and y = 6. - To find the variation equation, we start with z = kyx^3. Plugging in the values z = 96, x = 2, and y = 6, we get 96 = k * 6 * 2^3, which simplifies to 96 = 48k. Dividing both sides by 48, we find k = 2. So, the variation equation is z = 2yx^3. None of the given answer choices match this equation.

40. z varies jointly as x and y; z = 84 when x = 28 and y = 12. - Your equation z = xy is correct. To find the constant of variation, plug in the values z = 84, x = 28, and y = 12 into the equation. We get 84 = k * 28 * 12. Simplifying, 84 = 336k. Dividing both sides by 336, we find k = 1/4.

46. h varies jointly as f and g; f = 27, g = 12, and k = 3. - Your equation h = kfg is correct. Plugging in the values f = 27, g = 12, and k = 3 into the equation, we get h = 3 * 27 * 12, which simplifies to h = 972.

50. p varies directly as r and inversely as s. - Your equation p = kr/s is correct.

53. r varies directly as s and inversely as the square of t. - Your equation r = ks/t^2 is correct.

58. z varies directly as x and inversely as y; z = 2 when x = 30 and y = 60. - Your equation z = kx/y is correct. Plugging in the values z = 2, x = 30, and y = 60, we get 2 = k * 30/60. Simplifying, 2 = k/2. Multiplying both sides by 2, we find k = 4.

60. z varies directly as x and inversely as y; z = 4 when x = 77 and y = 28. - Your equation z = kx/y is correct. Plugging in the values z = 4, x = 77, and y = 28, we get 4 = k * 77/28. Simplifying, 4 = 77k/28. Multiplying both sides by 28, we find 112 = 77k. Dividing both sides by 77, we get k = 112/77, which can be reduced to k = 16/11.

66. y varies jointly as a and b and inversely as the square root of c; y = 24 when a = 6, b = 4, and c = 49. - Your equation y = kab/sqrt(c) is correct. Plugging in the values y = 24, a = 6, b = 4, and c = 49, we get 24 = k * 6 * 4/sqrt(49). Simplifying, 24 = 24k/7. Multiplying both sides by 7, we find 168 = 24k. Dividing both sides by 24, we get k = 7. So, the variation equation is y = 7ab/sqrt(c). To find y when a = 2, b = 5, and c = 16, just plug in these values into the equation: y = 7 * 2 * 5/sqrt(16). Simplifying, y = 70/4, which reduces to y = 17.5.

Overall, most of your answers are correct. Great job!