Sure, let's go through the steps in detail to solve this problem.
Step 1: Convert the race distance from miles to yards.
- 1 mile = 1760 yards
- Race distance in miles = 10 miles
- To convert to yards: 10 miles × 1760 yards/mile
So, the total race distance in yards is:
[tex]\[ 10 \times 1760 = 17600 \text{ yards} \][/tex]
Step 2: Determine the intervals at which the flags need to be placed.
- Flag distance: 275 yards
Step 3: Calculate the number of intervals (flags) needed to cover the entire race distance, not forgetting to place a flag at the finish line.
[tex]\[ \text{Number of flags} = \left\lfloor \frac{\text{Total race distance in yards}}{\text{Flag distance in yards}} \right\rfloor + 1 \][/tex]
Where:
- Total race distance in yards = 17600 yards
- Flag distance in yards = 275 yards
Substituting these values in, we get:
[tex]\[ \left\lfloor \frac{17600}{275} \right\rfloor + 1 \][/tex]
[tex]\[ = 64 + 1 \][/tex]
[tex]\[ = 65 \][/tex]
So, the number of flags needed is:
[tex]\[ 65 \][/tex]
To summarize, the racecourse is 17600 yards long, and the officials need to place 65 flags every 275 yards, including one at the finish line.
