Final answer:
An equation connecting w and z is z² = 2 * w. When w is 18, z can be ±6. When z is 5, w is 12.5. The graph of z² against w is a straight line through the origin with a slope of the constant of proportionality.
Step-by-step explanation:
Solution for a Directly Proportional Relationship
Given that z2 is directly proportional to w, and z = 4 when w = 8, we can establish a relationship between z and w.
Equation Connecting w and z
To find the equation connecting w and z, we use the principle of direct proportionality:
z2 = k * w
Where k is the constant of proportionality. Given z = 4 when w = 8:
42 = k * 8
16 = k * 8
k = 2
So, the equation that connects w and z is:
z2 = 2 * w
Values of z when w = 18
z2 = 2 * 18
z2 = 36
z = ±6 (Since a square root has both positive and negative roots)
Value of w when z = 5
z2 = 2 * w
52 = 2 * w
25 = 2 * w
w = 12.5
Graph of z2 against w
To draw the graph, we would plot z2 on the y-axis and w on the x-axis, resulting in a straight line that passes through the origin with a slope equal to the constant of proportionality, k. The line represents the equation z2 = 2*w.