Question
A glass bottle filled with oil weighs
\[590\] grams. After Sophia uses
\[200\] milliliters of oil, the bottle of oil weighs
\[400\] grams.
The weight
\[M\] of the bottle of oil, in grams, is a function of
\[V\], the volume, in milliliters, of oil Sophia has used.
Write the function's formula.
\[M=\]
\[590\] grams. After Sophia uses
\[200\] milliliters of oil, the bottle of oil weighs
\[400\] grams.
The weight
\[M\] of the bottle of oil, in grams, is a function of
\[V\], the volume, in milliliters, of oil Sophia has used.
Write the function's formula.
\[M=\]
Answers
GPT-4o mini
To find the equation that describes the weight \(M\) of the bottle of oil based on the volume \(V\) of oil that has been used, we start with the given information.
1. Initially, the total weight of the full bottle of oil is \(590\) grams, which includes the weight of the empty bottle and the weight of the full amount of oil.
2. After using \(200\) milliliters of oil, the bottle weighs \(400\) grams.
From this, we can derive the weight of the bottle and the weight of the oil.
### Step 1: Calculate the Weight of the Used Oil
The decrease in weight after using \(200\) milliliters of oil gives us the weight of the oil that was used:
\[
\text{Weight before using oil} - \text{Weight after using oil} = \text{Weight of used oil}
\]
\[
590 \text{ grams} - 400 \text{ grams} = 190 \text{ grams}
\]
This means that \(200\) milliliters of oil weighs \(190\) grams.
### Step 2: Calculate the Density of the Oil
We can find the weight of \(1\) milliliter of oil:
\[
\text{Weight per milliliter} = \frac{190 \text{ grams}}{200 \text{ milliliters}} = 0.95 \text{ grams/milliliter}
\]
### Step 3: Establish the Function
Let \( V \) be the volume of oil that has been used (in milliliters). The weight of oil that has been used can be calculated as:
\[
\text{Weight of used oil} = 0.95V \text{ grams}
\]
The weight of the bottle when \(V\) milliliters of oil has been used can be represented as:
\[
M = \text{Weight of full bottle} - \text{Weight of used oil}
\]
Substituting the values:
\[
M = 590 \text{ grams} - 0.95V
\]
Thus, the function for the weight \(M\) of the bottle filled with oil, in grams, as a function of the volume \(V\), in milliliters, of oil used is:
\[
\boxed{M = 590 - 0.95V}
\]
1. Initially, the total weight of the full bottle of oil is \(590\) grams, which includes the weight of the empty bottle and the weight of the full amount of oil.
2. After using \(200\) milliliters of oil, the bottle weighs \(400\) grams.
From this, we can derive the weight of the bottle and the weight of the oil.
### Step 1: Calculate the Weight of the Used Oil
The decrease in weight after using \(200\) milliliters of oil gives us the weight of the oil that was used:
\[
\text{Weight before using oil} - \text{Weight after using oil} = \text{Weight of used oil}
\]
\[
590 \text{ grams} - 400 \text{ grams} = 190 \text{ grams}
\]
This means that \(200\) milliliters of oil weighs \(190\) grams.
### Step 2: Calculate the Density of the Oil
We can find the weight of \(1\) milliliter of oil:
\[
\text{Weight per milliliter} = \frac{190 \text{ grams}}{200 \text{ milliliters}} = 0.95 \text{ grams/milliliter}
\]
### Step 3: Establish the Function
Let \( V \) be the volume of oil that has been used (in milliliters). The weight of oil that has been used can be calculated as:
\[
\text{Weight of used oil} = 0.95V \text{ grams}
\]
The weight of the bottle when \(V\) milliliters of oil has been used can be represented as:
\[
M = \text{Weight of full bottle} - \text{Weight of used oil}
\]
Substituting the values:
\[
M = 590 \text{ grams} - 0.95V
\]
Thus, the function for the weight \(M\) of the bottle filled with oil, in grams, as a function of the volume \(V\), in milliliters, of oil used is:
\[
\boxed{M = 590 - 0.95V}
\]