Sure, let's work through this problem step-by-step. Alan wants to install security cameras in his store, and we need to find out how many cameras will be necessary to cover the entire store and how much area would remain uncovered if he only places cameras in the corners.
### Step 1: Calculate the area of the store
The store has dimensions of 30 feet by 30 feet. To find the area, we use the formula for the area of a rectangle:
[tex]\[
\text{Area of the store} = \text{length} \times \text{width}
\][/tex]
[tex]\[
\text{Area of the store} = 30 \text{ ft} \times 30 \text{ ft} = 900 \text{ square feet}
\][/tex]
### Step 2: Determine the viewing area of each camera
Each camera can view an area of approximately [tex]\(112.5 \text{ square feet}\)[/tex].
### Step 3: Calculate the number of cameras needed to cover the entire store
We need to find out how many cameras are required to cover the store's total area of 900 square feet. We do this by dividing the total area of the store by the viewing area of one camera:
[tex]\[
\text{Number of cameras needed} = \frac{\text{Area of the store}}{\text{Viewing area of each camera}}
\][/tex]
[tex]\[
\text{Number of cameras needed} = \frac{900 \text{ square feet}}{112.5 \text{ square feet}} \approx 8
\][/tex]
Therefore, Alan will need 8 cameras to cover the entire store.
### Step 4: Calculate the total viewing area if cameras are placed only in the corners
If Alan places cameras only in the corners of the store, there will be 4 cameras placed at each corner. The total viewing area covered by these 4 cameras is:
[tex]\[
\text{Total viewing area from corner cameras} = 4 \times \text{Viewing area of each camera}
\][/tex]
[tex]\[
\text{Total viewing area from corner cameras} = 4 \times 112.5 \text{ square feet} = 450 \text{ square feet}
\][/tex]
### Step 5: Calculate the uncovered area
To find out the area of the store that is left uncovered when cameras are placed only in the corners, we subtract the total viewing area covered by the corner cameras from the total area of the store:
[tex]\[
\text{Uncovered area} = \text{Area of the store} - \text{Total viewing area from corner cameras}
\][/tex]
[tex]\[
\text{Uncovered area} = 900 \text{ square feet} - 450 \text{ square feet} = 450 \text{ square feet}
\][/tex]
### Conclusion
- Alan will need 8 cameras to cover the entire store.
- If Alan places cameras only in the corners of the store, 450 square feet of the store will remain uncovered.
