Polynomial word problem: rectangle and circle area

right 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 7 R 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 2 R so what it's what's its area well it's just going to be the length times the width so the area here the area here is going to be P maybe I should say to R P to R P this is the length times the width or the width times the length so area is equal to 2 R P for the rectangle now we also we want to find the difference between this area and the area of a circle 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 its diameter is 4 R how do we figure out the area of a circle area area is equal to PI R squared for a circle where R is the 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 there 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 area of the circle is equal to 4 PI R squared and then what we want to find the difference so before to find the difference it's helpful just so we don't end up with a negative number then figure out which of these two is larger so they're telling us that p is greater than that P is greater than 7 R so let's think about this if P is greater than 7 R then to let me write it this way we know that P is greater than 7 R so if we were to multiply sides of this equation by 2 R and 2 R 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 2r p is greater than 14 is greater than 14 sorry not 514 r-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 area of the rectangle and what's 14 r-squared well 4 times pi 4 times pi is going to get us something less than 14 this is less than 14 so this is 4 pi 4 pi is less than 14 14 is 4 times 3 & 2 it'll be get me make sure no 14 is less 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 the air this rectangle has a larger area than the circle so we can just subtract the circles area from the rectangles area to find the difference so the difference is going to be the area of the rectangle which we already figured out is 2 RP 2 RP and we're going to subtract from that the area of the circle the area of the circle is 4 PI 4 PI R squared so hopefully that made sense and one point I want to clarify gave the equation I gave the equation of a circle the area of a circle to be PI R squared and then we said that the radius is actually 2 R in this case so I substituted 2 R 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 two times some letter R so I substitute that into the formula anyway hopefully you found that useful