z varies jointly with y and the square of x. If x=-2 when y=7 and z=-84,find x when z=-96 and y=2
I just need help figuring out what equation to use.
Well, let me try to help you out with a little humor!
If z varies jointly with y and the square of x, that means they are having a "proportional party"!
To find the equation, we can say that:
z = k * y * x^2
Now, let's solve for the constant of proportionality, k.
When x = -2, y = 7, and z = -84, we can plug those values into the equation:
-84 = k * 7 * (-2)^2
Simplifying the equation, we get:
-84 = 4k * 7
Dividing both sides by 28, we find:
k = -3
Now we have our equation:
z = -3 * y * x^2
To find x when z = -96 and y = 2, we can plug in these values into the equation:
-96 = -3 * 2 * x^2
Simplifying, we get:
-96 = -6x^2
Dividing both sides by -6, we find:
x^2 = 16
Taking the square root of both sides, we get:
x = ±4
So, x can be either 4 or -4 when z = -96 and y = 2.
To find the equation that relates the variables x, y, and z, we can start by writing the general equation for joint variation:
z = k * x^m * y^n
where k is the constant of variation and m, n are the exponents.
Given that "z varies jointly with y and the square of x," we can rewrite the equation as:
z = k * x^2 * y^1
Now, let's find the value of k. We can use the first set of values provided:
x = -2, y = 7, z = -84
Substituting these values into the equation, we get:
-84 = k * (-2)^2 * 7^1
-84 = k * 4 * 7
-84 = 28k
Solving for k, we divide both sides by 28:
k = -84 / 28
k = -3
Now that we have the value of k, we can use it to find x when z = -96 and y = 2.
Substituting the new values into the equation, we get:
-96 = -3 * x^2 * 2^1
-96 = -6x^2
Dividing both sides by -6, we have:
16 = x^2
Taking the square root of both sides, we find:
x = ±√16
x = ±4
Therefore, when z = -96 and y = 2, x can be either -4 or +4.
To solve the given problem, you need to use the concept of joint variation. In joint variation, a variable varies directly with one factor and inversely with another.
In this case, z directly varies with y and the square of x. Mathematically, this can be expressed as:
z = k * x^2 * y
where k is a constant of variation.
To find the equation, you can plug in the given values when x = -2, y = 7, and z = -84. This will help you solve for the constant of variation, k.
-84 = k * (-2)^2 * 7
Simplifying this expression:
-84 = 4k * 7
-84 = 28k
Now, divide both sides of the equation by 28:
-84/28 = k
-3 = k
Therefore, the equation for the variation is:
z = -3x^2y
To find x when z = -96 and y = 2, you can substitute these values into the equation and solve for x:
-96 = -3x^2 * 2
Simplifying this expression:
-96 = -6x^2
Now, divide both sides of the equation by -6:
-96 / -6 = x^2
16 = x^2
Taking the square root of both sides, you get:
x = ± 4
So, when z = -96 and y = 2, there are two possible solutions for x: x = 4 or x = -4.
z = kyx^2
so,
z/(yx^2) = k is constant. So, you want x such that
-96/(2x^2) = -84/(7*4)