Math Problem Statement

Suppose y varies jointly with x and z. If y = 20 when x = 2 and z = 5, find the constant k. Then find y when x = 4 and z = 8.

Solution

From the image, the problem states:

"Suppose yy varies jointly with xx and zz, i.e.,
y=kxzy = kxz.

If y=20y = 20 when x=2x = 2 and z=5z = 5, find kk.

Then find yy when x=4x = 4 and z=8z = 8."


Step 1: Solve for kk

We use the equation y=kxzy = kxz. Plugging in the given values:

20=k2520 = k \cdot 2 \cdot 5

Simplify:

20=10kk=2010=220 = 10k \quad \Rightarrow \quad k = \frac{20}{10} = 2

Thus, k=2k = 2.


Step 2: Find yy when x=4x = 4 and z=8z = 8

Using k=2k = 2 and the equation y=kxzy = kxz:

y=248y = 2 \cdot 4 \cdot 8

Simplify:

y=64y = 64


Final Answer:

  • k=2k = 2
  • y=64y = 64 when x=4x = 4 and z=8z = 8.

Would you like a detailed explanation of any step?


Related Questions:

  1. What is joint variation, and how does it differ from direct variation?
  2. How can you determine the constant of proportionality (kk) in other joint variation problems?
  3. If y=kxzy = kxz and k=3k = 3, what is yy when x=6x = 6 and z=7z = 7?
  4. How can the formula for joint variation be modified for inverse relationships?
  5. What happens to yy if xx doubles and zz remains the same?

Tip:

Always isolate the constant kk first in variation problems—it helps simplify the rest of the solution!

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Math Problem Analysis

Mathematical Concepts

Joint Variation
Algebra

Formulas

y = kxz

Theorems

Variation Principle

Suitable Grade Level

Grades 8-10