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Explain 1: Rearranging Scientific Formulas

Use inverse operations to isolate the unknown variable in a scientific formula.

The formula for density is [tex]\( D = \frac{m}{V} \)[/tex]. Lead has a very high density of [tex]\( 11,340 \, kg/m^3 \)[/tex]. Plastic foam has a very low density of [tex]\( 75 \, kg/m^3 \)[/tex]. The formula for density can be rearranged to solve for [tex]\( V \)[/tex] (volume) or [tex]\( m \)[/tex] (mass).

Example 1:

(A) A sinker on a fishing line is made of lead and has a volume of [tex]\( 0.000015 \, m^3 \)[/tex]. What is the mass of the sinker?

The density formula can be rearranged to isolate [tex]\( m \)[/tex], the mass. The values for volume and density can then be substituted into the formula to find the mass.

[tex]\[
D \cdot V = m
\][/tex]

[tex]\[
(11,340 \, kg/m^3) \cdot (0.000015 \, m^3) = m
\][/tex]

[tex]\[
0.17 \, kg \approx m
\][/tex]

(B) The design for a life preserver requires 0.3 kilograms of plastic foam to provide proper buoyancy. What is the volume of the plastic foam required?

Rearrange the density formula to isolate [tex]\( V \)[/tex]:

[tex]\[
D = \frac{m}{V}
\][/tex]

[tex]\[
D \cdot V = m
\][/tex]

[tex]\[
V = \frac{m}{D}
\][/tex]

Now substitute the given values:

[tex]\[
V = \frac{0.3 \, kg}{75 \, kg/m^3}
\][/tex]

[tex]\[
V = 0.004 \, m^3
\][/tex]



Answer :

GinnyAnswer
Sure, let's go through the process of rearranging the scientific formula for density and solving both parts of the problem step-by-step.

### Understanding the Density Formula
The general formula for density [tex]\(D\)[/tex] is expressed as:
[tex]\[ D = \frac{m}{V} \][/tex]
where:
- [tex]\(D\)[/tex] is the density in [tex]\(kg/m^3\)[/tex],
- [tex]\(m\)[/tex] is the mass in kilograms (kg),
- [tex]\(V\)[/tex] is the volume in cubic meters ([tex]\(m^3\)[/tex]).

This formula can be rearranged to solve for either [tex]\(m\)[/tex] (mass) or [tex]\(V\)[/tex] (volume).

### Example 1 - Part (A)
Question: A sinker on a fishing line is made of lead and has a volume of [tex]\(0.000015 m^3\)[/tex]. What is the mass of the sinker?

Given:
- Density of lead ([tex]\(D_{\text{lead}}\)[/tex]) = 11340 [tex]\(kg/m^3\)[/tex]
- Volume of lead ([tex]\(V_{\text{lead}}\)[/tex]) = 0.000015 [tex]\(m^3\)[/tex]

Step-by-step solution:
1. Start with the density formula:
[tex]\[ D = \frac{m}{V} \][/tex]
2. Rearrange the formula to solve for mass ([tex]\(m\)[/tex]):
[tex]\[ m = D \times V \][/tex]
3. Substitute the given values into the formula:
[tex]\[ m = 11340 \, \text{kg/m}^3 \times 0.000015 \, \text{m}^3 \][/tex]
4. Perform the multiplication:
[tex]\[ m = 0.1701 \, \text{kg} \][/tex]

So, the mass of the lead sinker is approximately [tex]\(0.1701\)[/tex] kilograms.

### Example 1 - Part (B)
Question: The design for a life preserver requires 0.3 kilograms of plastic foam to provide proper buoyancy. What is the volume of the plastic foam required?

Given:
- Density of plastic foam ([tex]\(D_{\text{foam}}\)[/tex]) = 75 [tex]\(kg/m^3\)[/tex]
- Mass of plastic foam ([tex]\(m_{\text{foam}}\)[/tex]) = 0.3 [tex]\(kg\)[/tex]

Step-by-step solution:
1. Start with the density formula:
[tex]\[ D = \frac{m}{V} \][/tex]
2. Rearrange the formula to solve for volume ([tex]\(V\)[/tex]):
[tex]\[ V = \frac{m}{D} \][/tex]
3. Substitute the given values into the formula:
[tex]\[ V = \frac{0.3 \, \text{kg}}{75 \, \text{kg/m}^3} \][/tex]
4. Perform the division:
[tex]\[ V = 0.004 \, \text{m}^3 \][/tex]

So, the volume of the plastic foam required is [tex]\(0.004\)[/tex] cubic meters.

### Summary
(A) The mass of the lead sinker is [tex]\(0.1701\)[/tex] kilograms.
(B) The volume of the plastic foam required for the life preserver is [tex]\(0.004\)[/tex] cubic meters.

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