z varies jointly as the square of x and the 5th power of y.
z = k x^2 y^5
Ah, so z is practically a drama queen who loves the attention from x and y. It's like a twisted love triangle, you know? Like x and y are fighting for z's affections, but instead of flowers and chocolates, they're battling it out with math equations. So, z = kx^2y^5, where k is the constant that tries to keep the peace between x and y. It's quite the soap opera, I must say!
The mathematical statement "z varies jointly as the square of x and the 5th power of y" can be written as:
z = k * x^2 * y^5
Here, z represents the variable that varies, x and y are the independent variables, and k is the constant of variation. This equation shows that as x and y change, z changes proportionally. The constant of variation, k, determines the specific relationship between z, x, and y.
To determine how z varies jointly as the square of x and the 5th power of y, we can write the proportionality equation as:
z = k * x^2 * y^5
Here, k is the constant of variation.
To find the constant of variation, we need additional information. This could be expressed in the form of an equation or values for z, x, and y.
For example, if we are given that when x = 2, y = 3, and z = 24, we can substitute these values into the equation to solve for k:
24 = k * (2)^2 * (3)^5
Using exponentiation:
24 = k * 4 * 243
Rearranging the equation:
24 = 972k
Dividing both sides by 972:
k = 24/972 = 1/40.5 ≈ 0.0247
Now that we have the constant of variation, we can express the relationship:
z = 0.0247 * x^2 * y^5
Therefore, z varies jointly as the square of x and the 5th power of y with a constant of variation approximately equal to 0.0247.