I do not know where to start with this one. Suppose p varies directly as the square of z and inversely as r. If p = 32/5 when z = 4 and r = 10, find p when z = 3 and r = 36.
" p varies directly as the square of z and inversely as r "
----> p = k (√z) (1/r) , where k is a constant
given: p=32/5 , z=4, r=10
32/5 = k(√4)(1/10)
times 10
64 = 2k
k = 32
so p = 32√z/r
then when z=3,r=36
p = 32√3/36
= 8√3/9 or appr 1.54
6x+4(70-x)=300
Step 1: Understand the given information and problem statement.
The problem states that p varies directly as the square of z and inversely as r. This means that when z increases or decreases, p will also increase or decrease, and when r increases or decreases, p will have the opposite effect.
We are given that when z = 4 and r = 10, p = 32/5. We need to find the value of p when z = 3 and r = 36.
Step 2: Write the equation that represents the direct and inverse variation.
The equation for direct variation is p = kz^2, where k is the constant of variation.
The equation for inverse variation is p = d/r, where d is another constant of variation.
Combining both equations, we get p = k(z^2)/r.
Step 3: Use the given values to find the value of the constant of variation k.
Given:
p = 32/5
z = 4
r = 10
Substituting these values into the equation p = k(z^2)/r:
32/5 = k(4^2)/10
32/5 = 16k/10
32/5 = 8k/5
Cross multiplying, we get:
5(8k) = 32(5)
40k = 160
k = 160/40
k = 4
Step 4: Solve for p when z = 3 and r = 36.
Given:
z = 3
r = 36
k = 4 (from Step 3)
Substituting these values into the equation p = k(z^2)/r:
p = 4(3^2)/36
p = 4(9)/36
p = 36/36
p = 1
So, when z = 3 and r = 36, p = 1.
To solve this problem, let's break it down step by step.
1) We are given that p varies directly as the square of z and inversely as r. This can be represented as:
p = k * (z^2) / r
where k is the constant of variation.
2) Next, we are given that p = 32/5 when z = 4 and r = 10. We can substitute these values into the equation and solve for k:
32/5 = k * (4^2) / 10
32/5 = 16k / 10
32/5 = 8k / 5
Multiplying both sides by 5:
32 = 8k
Dividing both sides by 8:
4 = k
So, k = 4.
3) Now that we have the value of k, we can find p when z = 3 and r = 36. Substitute these values into the equation:
p = 4 * (3^2) / 36
p = 4 * 9 / 36
p = 36 / 36
p = 1
Therefore, when z = 3 and r = 36, p is equal to 1.