If x varies directly as y62, inversely as z, and inversely as the square root of w, what happens to x when y is multiplied by 3, z is doubled, and w is quadrupled?
I would really appriciate it if someone could help me understand how to do this problem. Thank you!
To solve this problem, we need to understand the concept of direct variation, inverse variation, and inverse variation with the square root.
1. Direct Variation: When two variables are directly proportional, it means that as one variable increases, the other variable increases by the same factor. In mathematical terms, it can be represented as x = k * y, where k is the constant of variation.
2. Inverse Variation: When two variables are inversely proportional, it means that as one variable increases, the other variable decreases by the same factor. Mathematically, it can be represented as x = k / z, where k is the constant of variation.
3. Inverse Variation with Square Root: When one variable varies inversely with the square root of another variable, it means that as one variable increases, the other variable decreases by the square root of that increase. Mathematically, it can be represented as x = k / √w, where k is the constant of variation.
Now, let's analyze the problem step by step.
Given that x varies directly as y^62, inversely as z, and inversely as the square root of w, we can express this relationship mathematically as follows:
x = k * (y^62) / z * 1 / √w
To find out what happens to x when y is multiplied by 3, z is doubled, and w is quadrupled, we'll examine the impact on each variable one at a time.
1. When y is multiplied by 3:
To account for this change, we need to substitute y with 3y in our equation:
x = k * ((3y)^62) / z * 1 / √w
2. When z is doubled:
To account for this change, we need to multiply z by 2 in our equation:
x = k * ((3y)^62) / (2z) * 1 / √w
3. When w is quadrupled:
To account for this change, we need to substitute w with (4w) in our equation:
x = k * ((3y)^62) / (2z) * 1 / √(4w)
Now, simplifying the equation further:
x = k * (3^62 * y^62) / (2z) * 1 / (2√w)
x = k * (3^62 * y^62) / (4z) * 1 / √w
From the final equation, we can observe that x is multiplied by (3^62 * y^62) and divided by both (4z) and √w.
Therefore, when y is multiplied by 3, z is doubled, and w is quadrupled, x will be multiplied by (3^62 * y^62) and divided by (4z) and √w.