The javelin event is held on the field inside the track. This field is a rectangle measuring 100 meters by 64 meters, with a semicircle at each end.
Early modern Olympic javelins were designed so well that javelin throwers could throw them beyond the field and onto the track. This was obviously dangerous for runners! Possible solutions to this problem included redesigning the javelin so it wouldn't "float" so much and removing the 2 innermost track lanes.
If the area of the grass, before removing the innermost track lanes, is approximately 9615 square meters, how many square kilometers of grass are there?
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To solve this problem, we need to calculate the area of the grass in square kilometers.
First, let's find the total area of the field, including the two semi-circular regions at the ends. The field is a rectangle measuring 100 meters by 64 meters, so the area of the rectangle is:
Area of rectangle = Length x Width = 100m x 64m = 6400 square meters
Next, let's calculate the area of one semi-circle. A semi-circle is half of a circle, so the formula to calculate the area of a semi-circle is:
Area of semi-circle = (π x radius²) / 2
The diameter of the semi-circle is the same as the width of the rectangle, which is 64 meters. So the radius of the semi-circle is half of the width, which is 32 meters. Substituting these values into the formula, we get:
Area of one semi-circle = (π x 32m²) / 2
Now, let's calculate the total area of both semi-circles:
Total area of the two semi-circles = 2 x (π x 32m²) / 2 = π x 32m²
Adding the area of the rectangle and the total area of the semi-circles will give us the total area of the field:
Total area of the field = Area of rectangle + Total area of the two semi-circles
= 6400 square meters + π x 32m²
Next, we know that the area of the grass before removing the innermost track lanes is approximately 9615 square meters. So we can set up an equation:
6400 square meters + π x 32m² = 9615 square meters
To find the value of π, you can use an approximation such as 3.14.
Substituting the values into the equation:
6400 + π x 32² = 9615
Now, we can solve the equation by rearranging it to isolate the term with π:
π x 32² = 9615 - 6400
π x 32² = 3215
Finally, divide both sides of the equation by 32² to solve for the value of π:
π = 3215 / (32²)
Now that we have the value of π, we can substitute it back into the equation for the total area of the field to find the area in square kilometers. Divide the total area of the field in square meters by 1,000,000 (as there are 1,000,000 square meters in a square kilometer) to get the area in square kilometers.