Alright, let's solve this step-by-step to determine the approximate cost for the banner.
### Step 1: Determine the dimensions of the parallelogram
Jace ordered a banner with the following dimensions:
- [tex]\(a = 7\)[/tex] feet
- [tex]\(b = 8\)[/tex] feet
- [tex]\(c = 9\)[/tex] feet
### Step 2: Calculate the semi-perimeter
The semi-perimeter [tex]\(s\)[/tex] of the parallelogram is calculated by adding the lengths of the sides and dividing by 2:
[tex]\[ s = \frac{a + b + c}{2} \][/tex]
Substituting the given values:
[tex]\[ s = \frac{7 + 8 + 9}{2} \][/tex]
[tex]\[ s = \frac{24}{2} \][/tex]
[tex]\[ s = 12 \text{ feet} \][/tex]
### Step 3: Calculate the area using Heron's formula
Heron's formula for the area of a triangle is given by:
[tex]\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \][/tex]
Substituting the semi-perimeter and side lengths:
[tex]\[ \text{Area} = \sqrt{12 \times (12 - 7) \times (12 - 8) \times (12 - 9)} \][/tex]
[tex]\[ \text{Area} = \sqrt{12 \times 5 \times 4 \times 3} \][/tex]
[tex]\[ \text{Area} = \sqrt{720} \][/tex]
[tex]\[ \text{Area} \approx 26.83 \text{ square feet} \][/tex]
### Step 4: Calculate the cost of the banner
The print shop charges [tex]$1.10 per square foot for the banner.
So, the cost \(C\) can be calculated as:
\[ C = \text{Area} \times \text{cost per square foot} \]
\[ C = 26.83 \times 1.10 \]
\[ C \approx 29.52 \]
### Step 5: Compare the calculated cost with the provided options
The calculated cost of the banner is approximately $[/tex]29.52 before tax. The provided options for the banner are:
- [tex]$41.95
- $[/tex]46.14
- [tex]$83.90
- $[/tex]92.30
None of the provided options match the calculated cost.
Therefore, verifying the calculations provides the conclusion that the cost for the banner before tax is:
[tex]\[ \$ 29.52 \][/tex]
Hence, the closest approximation should be considered as the direct result.
