To determine the pressure needed when the distance from the object is 3 inches, given that the pressure varies inversely as the distance, we can use the relationship of inverse proportionality.
1. Understand Inverse Proportionality:
When two quantities are inversely proportional, the product of the two quantities remains constant. In this case:
[tex]\[
\text{pressure} \times \text{distance} = \text{constant}
\][/tex]
2. Given Data:
We are told that 150 pounds of pressure is required when the distance is 10 inches.
Let's denote the initial pressure as \( P_1 \) and the initial distance as \( D_1 \):
[tex]\[
P_1 = 150 \text{ pounds}, \quad D_1 = 10 \text{ inches}
\][/tex]
3. Set Up the Relationship:
From the proportionality statement, we set up the relationship:
[tex]\[
P_1 \times D_1 = \text{constant}
\][/tex]
Plugging in the given values:
[tex]\[
150 \times 10 = \text{constant}
\][/tex]
[tex]\[
\text{constant} = 1500
\][/tex]
4. Find the Required Pressure:
Let's denote the required pressure as \( P_2 \) and the new distance as \( D_2 \):
[tex]\[
D_2 = 3 \text{ inches}
\][/tex]
Using the inverse proportionality, we establish the equation:
[tex]\[
P_2 \times D_2 = \text{constant}
\][/tex]
We already know the constant is 1500:
[tex]\[
P_2 \times 3 = 1500
\][/tex]
5. Solve for \( P_2 \):
[tex]\[
P_2 = \frac{1500}{3}
\][/tex]
6. Calculate the Result:
[tex]\[
P_2 = 500 \text{ pounds}
\][/tex]
Therefore, the pressure needed for a distance of 3 inches from the object is [tex]\( 500 \)[/tex] pounds.
