Suppose you are given that z = c x2 y4 where c is some number. For a particular value of c, x and y, z is equal to 7. Suppose x is decreased by a factor of 2.6 and y is increased by a factor 4. What is the new value of z?
Z = 7 * (1/2.6) * 4 = 10.77.
Correction: Z = 7 * (1/2.6)^2 * 4^4 = 265.1
To find the new value of z after making changes to x and y, we need to substitute the new values of x and y into the equation z = c * x^2 * y^4.
Let's break down the problem step-by-step:
1. Start with the initial equation: z = c * x^2 * y^4.
2. Given that z is equal to 7 for a particular value of c, x, and y.
Substitute z = 7 into the equation: 7 = c * x^2 * y^4.
3. Now, we need to make the changes to x and y:
- Decrease x by a factor of 2.6: new_x = x / 2.6.
- Increase y by a factor of 4: new_y = y * 4.
4. Substitute the new values into the equation: 7 = c * (new_x)^2 * (new_y)^4.
Replace new_x and new_y: 7 = c * (x / 2.6)^2 * (y * 4)^4.
5. Simplify the equation:
- (x / 2.6)^2 = x^2 / (2.6^2) = x^2 / 6.76.
- (y * 4)^4 = y^4 * 4^4 = y^4 * 256.
6. Substitute the simplifications back into the equation: 7 = c * (x^2 / 6.76) * (y^4 * 256).
Simplify further: 7 = (c * 256 / 6.76) * (x^2 * y^4).
Let k = (c * 256 / 6.76) be the new constant.
7. Finally, the equation becomes: 7 = k * (x^2 * y^4).
Rearrange the equation to find the new value of z:
z = 7 / k = 7 / (c * 256 / 6.76) = 1.059 * (6.76 / (c * 256)).
So, the new value of z, after decreasing x by a factor of 2.6 and increasing y by a factor of 4, is approximately 1.059 * (6.76 / (c * 256)).