Mathematics: Find the area of the region bounded by the parabola y=5x2y=5x2, the tangent line to this parabola at (2,20)(2,20) and the xx axis.

Hello!The answer is:The total fall in 30 feet will be 7.5 inches.Why?To calculate the total fall in 30 feet, given the slope (decreasing rate for this case), we need to use the following equation:We are asked to calculate the total fall in 30 feet, so, substituting it into the equation, we have:Hence, we have that the total fall in 30 feet will be 7.5 inches.Have a nice day!...

Find the area of the region bounded by the parabola

Ответ:

AnaiyaKirksey8

12.03.2022

Mathematics

1. Find the equation of tangent line at point (2,20).

$y'=(5x^2)'=5\cdot 2x=10x,\\ \\y'(2)=10\cdot 2=20.$

The equation of the tangent line is

$y=20(x-2)+20,\\ \\y=20x-20.$

2. Express x:

$y=20x-20\Rightarrow x=\dfrac{y}{20}+1;\\ \\y=5x^2\Rightarrow x=\sqrt{\dfrac{y}{5}}.$

3. Find the area of bounded region:

$A=\int\limits^{20}_0 {\left(\dfrac{y}{20}+1-\sqrt{\dfrac{y}{5}} } \right)\, dy=\left(\dfrac{y^2}{40}+y-\dfrac{2\sqrt{y^3} }{3\sqrt{5} } \right)\big|^{20}_0=$