2. z varies jointly as x and y. If z= 3 when x = 3 and y = 15, find z when x = 6 and y = 9. 3. z varies jointly as the square root of the product of x and y. If z=3 when x = 3 and y = 12, find x when z= 6 and y = 64.
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Answer:
2. Given:
z varies jointly as x and y.
z = 3 when x = 3 and y = 15.
To find z when x = 6 and y = 9, we can set up a proportion using the joint variation equation:
z₁ / (x₁ * y₁) = z₂ / (x₂ * y₂)
Substituting the given values:
3 / (3 * 15) = z / (6 * 9)
Simplifying:
1 / 15 = z / 54
Cross-multiplying:
54 = 15z
Dividing both sides by 15:
z = 54 / 15
Simplifying:
z = 3.6
Therefore, when x = 6 and y = 9, z is equal to 3.6.
3. Given:
z varies jointly as the square root of the product of x and y.
z = 3 when x = 3 and y = 12.
To find x when z = 6 and y = 64, we can set up a proportion using the joint variation equation:
z₁ / √(x₁ * y₁) = z₂ / √(x₂ * y₂)
Substituting the given values:
3 / √(3 * 12) = 6 / √(x * 64)
Simplifying:
3 / √36 = 6 / (8√x)
Cross-multiplying:
3 * 8√x = 6 * √36
24√x = 6 * 6
Simplifying:
24√x = 36
Dividing both sides by 24:
√x = 36 / 24
Simplifying:
√x = 3/2
Squaring both sides:
x = (3/2)²
Simplifying:
x = 9/4
Therefore, when z = 6 and y = 64, x is equal to 9/4.
Step-by-step explanation:
In a short straight, the relationship between variables is described as a direct or proportional variation. This means that as one variable increases or decreases, the other variable also increases or decreases in a consistent manner.
For example, if we have the equation y = kx, where y and x are variables and k is a constant, this represents a short straight. It means that as x increases, y also increases proportionally. Similarly, if x decreases, y decreases proportionally.
The constant k represents the ratio of change between the variables. It determines the steepness of the line in a graph representing the short straight. A larger value of k indicates a steeper line, while a smaller value of k indicates a less steep line.
In summary, a short straight represents a direct or proportional relationship between variables, where one variable increases or decreases in proportion to the other variable.
Answer:
What kind of solution is that? ↑
Here is a precise solution:
1. z varies jointly as x and y:
- Given z = 3 when x = 3 and y = 15.
- To find z when x = 6 and y = 9, we can set up a proportion based on the joint variation:
z₁ / z₂ = (x₁ * y₁) / (x₂ * y₂)
Plug in the given values: 3 / z₂ = (3 * 15) / (6 * 9)
Simplifying, we get: 3 / z₂ = 45 / 54
Cross multiply: 45z₂ = 3 * 54
Divide both sides by 45: z₂ = (3 * 54) / 45
Simplifying gives us: z₂ = 3.6
2. z varies jointly as the square root of the product of x and y:
- Given z = 3 when x = 3 and y = 12.
- To find x when z = 6 and y = 64, we again set up a proportion based on the joint variation:
z₁ / z₂ = √(x₁ * y₁) / √(x₂ * y₂)
Substitute the given values: 3 / 6 = √(3 * 12) / √(x₂ * 64)
Simplify the square roots: 3 / 6 = √(36) / √(64x₂)
Further simplify: 3 / 6 = 6 / (8√(x₂))
Cross multiply: 3 * 8√(x₂) = 6 * 6
Simplify gives us: 24√(x₂) = 36
Divide both sides by 24: √(x₂) = 36 / 24
Square both sides: x₂ = (36 / 24)²
Calculate: x₂ = 1.5²
Finally, x₂ = 2.25
So, when x = 6 and y = 9, z is approximately 3.6, and when z = 6 and y = 64, x is approximately 2.25.