Polynomials word problems
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Write a binomial to express the difference between the area of a rectangle with length p and width 2r and the area of a circle with diameter 4r. And they tell us that p is greater than 7r. So let's first think about the area of a rectangle with length p and width 2r. So this is our rectangle right here. It has a length of p and it has a width of 2r. So what's its area? Well, it's just going to be the length times the width. So the area here is going to be p-- or maybe I should say 2rp. This is the length times the width, or the width times the length. So area is equal to 2rp for the rectangle. Now, we also want to find the difference between this area and the area of a circle with diameter 4r. So what's the area of the circle going to be? So let me draw our circle over here. So our circle looks like that. Its diameter is 4r. How do we figure out the area of a circle? Well, area is equal to pi r squared for a circle, where r is a radius. They gave us the diameter. The radius is half of that. So the radius here is going to be half this distance, or 2r. So the area of our circle is going to be pi times 2r, the whole thing squared. This is the radius, right? So we're squaring the entire radius. So this is going to be equal to pi times 4 times r squared. I'm just squaring each of these terms. Or if we were to change the order, the area of the circle is equal to 4 pi r squared. And we want to find the difference. So to find a difference, It's helpful-- just so we don't end up with a negative number-- to figure out which of these two is larger. So they're telling us that p is greater than 7 r. So let's think about this. If p is greater than 7r, then 2-- let me write it this way. We know that p is greater than 7r. So if we're going to multiply both sides of this equation by 2rr-- and 2r is positive, we're dealing with positive distances, positive lengths-- so if we multiply both sides of this equation by 2r, it shouldn't change the equation. So multiply that by 2r, and then multiply this by 2r. And then our equation becomes 2rp is greater than 14r squared. Now, why is this interesting? Actually, why did I even multiply this by 2r? Well, that's so that this becomes the same as the area of the rectangle. So this is the area of the rectangle. And what's 14r squared? Well, 4 times pi, is going to get us something less than 14. This is less than 14. So this is 4 pi is less than 14. 14 is 4 times 3 and 2-- let me put it this way. 4 times 3.5 is equal to 14. Right? So 4 times pi, which is less than 3.5, is going to be less than 14. So we know that this over here is larger than this quantity over here. It's larger than 4 pi r squared. And so we know that this rectangle has a larger area than the circle. So we can just subtract the circle's area from the rectangle's area to find the difference. So the difference is going to be the area of the rectangle, which we already figured out is 2rp. And we're going to subtract from that the area of the circle. The area of the circle is 4 pi r squared. So hopefully that made sense. And one point I want to clarify. I gave the equation of the area of a circle to be pi r squared. And then we said that the radius is actually 2r in this case. So I substituted 2r for r. Hopefully that doesn't confuse you. This r is the general term for any radius. They later told us that the actual radius is 2 times some letter r. So I substitute that into the formula. Anyway, hopefully you found that useful.