To find the cost of making the dress as a function of time, we can model the cost using a linear equation of the form:
\[ C = a + bT \]
where:
- \( C \) is the total cost,
- \( a \) is the constant cost (fixed part),
- \( b \) is the variable cost per hour,
- \( T \) is the time in hours.
We are given two points:
- When \( T = 3 \), \( C = 2700 \).
- When \( T = 5 \), \( C = 3100 \).
We can set up the following equations based on these points:
- \( 2700 = a + 3b \) (Equation 1)
- \( 3100 = a + 5b \) (Equation 2)
Now, we can solve these equations simultaneously. First, let's subtract Equation 1 from Equation 2:
\[ 3100 - 2700 = (a + 5b) - (a + 3b) \] \[ 400 = 2b \] \[ b = 200 \]
Now that we have \( b \), we can substitute it back into Equation 1 to find \( a \):
\[ 2700 = a + 3(200) \] \[ 2700 = a + 600 \] \[ a = 2700 - 600 \] \[ a = 2100 \]
Now we have:
- \( a = 2100 \)
- \( b = 200 \)
Thus, the cost function becomes:
\[ C = 2100 + 200T \]
Now we can find the cost if it takes \( 1.5 \) hours to make the dress by substituting \( T = 1.5 \) into the cost function:
\[ C = 2100 + 200(1.5) \] \[ C = 2100 + 300 \] \[ C = 2400 \]
Therefore, the cost of making the dress if it takes \( 1.5 \) hours is #2,400.
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