Question

z varies jointly as x and y. translate into variation statement. if z = 36 when x = 3 and y = 2, find k. x is 4 when y = 3 and z = 2. what is z if x
the area a of a triangle varies jointly as the base b and the height h. express the relation as a variation statement. if the area is 15 cm² when the base is 5 cm and the height is 6 cm, find k.

Explanation:

Step1: Write the joint - variation equation

The statement "z varies jointly as x and y" can be written as the equation $z = kxy$, where k is the constant of variation.

Step2: Find the value of k when z = 36, x = 3 and y = 2

Substitute the values into the equation $z = kxy$:
$36=k\times3\times2$.
Simplify the right - hand side: $36 = 6k$.
Solve for k by dividing both sides by 6: $k=\frac{36}{6}=6$.

Step3: Given x = 4, y = 3, k = 6, find z

Use the equation $z = kxy$.
Substitute k = 6, x = 4 and y = 3 into the equation: $z=6\times4\times3$.
Calculate the right - hand side: $z = 72$.

For the triangle problem:

Step4: Write the joint - variation equation for the triangle

The area a of a triangle varies jointly as the base b and the height h, so the equation is $a = kbh$.

Step5: Find the value of k when a = 15, b = 5 and h = 6

Substitute the values into the equation $a = kbh$:
$15=k\times5\times6$.
Simplify the right - hand side: $15 = 30k$.
Solve for k by dividing both sides by 30: $k=\frac{15}{30}=\frac{1}{2}$.

Answer:

For the first part: The variation statement is $z = kxy$, $k = 6$, and when $x = 4,y = 3$, $z = 72$.
For the triangle part: The variation statement is $a = kbh$, and $k=\frac{1}{2}$.