SPORTS In the diagram shown, a track surrounds a football field. How many bags of fertilizer wou

SPORTS In the diagram shown, a track surrounds a football...

Key Concepts

Composite Figures

Composite figures involve shapes that are constructed from a combination of simpler geometric figures. Understanding how to decompose or combine these figures is crucial for calculating the total area of irregular shapes, such as the grass area inside a track that surrounds another shape.

Area Calculation

Area calculation is a fundamental concept where formulas are used to compute the space contained within geometric shapes like rectangles, circles, and annuli. In geometric problems involving a surrounding track and an inner field, calculating the area typically involves determining the areas of the inner and outer figures and then finding the appropriate difference or sum.

Rounding and Unit Coverage

When applying a material based on unit coverage, it is essential to divide the total measured area by the area covered by one unit (in this case, a bag of fertilizer) and round up to ensure complete coverage. This accounts for partial units, ensuring that the entire region is adequately treated even if the divisions do not result in whole numbers.

Transcript

00:01 In this problem, what you're going to be doing is you're going to be trying to figure out how many bags of fertilizer you need to buy to fertilize this entire football field, basically.

00:11 Okay? and they say that each bag of fertilizer covers about 555 square yards.

00:17 And so let's investigate it.

00:19 You can see that i've marked in red the two semicircles that are the portion of the composite figure.

00:24 That's because together they add up to one circle, right? so the area of a circle is pire square.

00:29 So if you take this semicircle right on the left side and you add it to this semicircle on the right hand side in that yellow composite figure you're going to get a one value of one full circle right half pi r square plus half pi r square which is one whole pi r square so the first part of this is going to figure out what is the area of the full circle is the radius 25 yards so you do pi r squared where the radius is 25 so pi times 25 squared 25 squared is 625 so that would be 625 times pi, and that would be in square yards.

01:06 But we want to approximate it so we can combine it with the rectangle.

01:09 So we approximate pie at 3 .14.

01:12 So we get 625 times the approximate value of pi, 3 .14.

01:17 And that would give us, i'm just going to use a calculator to make it a little easier.

01:22 So if you took 625 and you multiplied it on a calculator by 3 .14, you would get 1 ,000.

01:31 962 .5.

01:32 So this is 1 ,962 .62 .5 square yards.

01:41 And that's the area of just the red that you see there, just the two semicircles added together.

01:46 But the composite figure also has this kind of green section here that i'm going to make, which is the rectangle or the main part of the football field, right? and that is 100 yards by 50 yards, right? so the base is going to be 100 yards and the height is going to it'd be 50 yards.

02:03 So the area of a rectangle is just base times height.

02:05 So you're just going to do base times height where the base in this case like we said is 100 and the height is 50.

02:12 And that's the same as 500 with a zero in the end.

02:15 So that's 5 ,000.

02:17 So you have 5 ,000 square yards for the area of the rectangle.

02:22 You have the area of the circles together, semi -circles together, which is going to be 1 ,9602 .5...

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