Mathematics, 31.08.2019 17:30 Courtneymorris19

Find the area of the region bounded by the parabola y=2x^2 , the tangent line to the parabola at (5,50), and the x-axis

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Check the picture below.

so.. the graph looks like so, since the tangent line is at 5,50, it touches the parabola when y = 50, x = 5.

so, let's get the tangent line at that point,

$\bf y=2x^2\implies \left. \cfrac{dy}{dx}=4x \right|_{x=5}\implies 20\leftarrow m
\\\\\\
y-y_1=m(x-x_1)\implies y-50=20(x-5)\implies y=20x-50\\
\left. \qquad \right. \uparrow\\
\textit{point-slope form}$

since I don't see an Up/Down function, just a Right/Left one... so, we'll use the function in "y-terms" then, the second one in the graph for each.

now... to get the bounds.... we could make them equal to each other, however, we know that the x-axis( y = 0) is the boundary line at the bottom, and that the tangent line is touching the parabolat a y = 50, thus, our bounds are from 0 to 50.

$\bf \begin{cases}
\cfrac{y+50}{20}=x\impliedby \textit{right function}\\\\
\sqrt{\cfrac{y}{2}}=x\impliedby \textit{left function}
\end{cases}
\\\\\\
\displaystyle \int\limits_{0}^{50}~\left[ \left( \cfrac{y+50}{20} \right)~-~\left( \sqrt{\cfrac{y}{2}}\right) \right]dy
\\\\\\
\displaystyle \cfrac{1}{20}\int\limits_{0}^{50}~y\cdot dy+\int\limits_{0}^{50}~\cfrac{5}{2}\cdot dy-\cfrac{1}{\sqrt{2}}\int\limits_{0}^{50}~y^{\frac{1}{2}}\cdot dy$

$\bf \left. \cfrac{y^2}{40}+\cfrac{5y}{2}-\cfrac{2\sqrt{y^3}}{3\sqrt{2}} \right]_{0}^{50}\implies \left[ \cfrac{250}{4}+125-\cfrac{500\sqrt{2}}{3\sqrt{2}} \right]-[0]
\\\\\\
\cfrac{375}{2}-\cfrac{500}{3}\implies \cfrac{125}{6}$

Find the area of the region bounded by the parabola y=2x^2 , the tangent line to the parabola at (5,

Find the area of the region bounded by the parabola y=2x^2 , the tangent line to the parabola at (5,

Answer from: Quest

the largest number of the four options is 1.8 x 10-2

you have it in scientific notation.

1.80 x 10-6 = 0.00000180

1.80 x 10-4 = 0.000180 (bigger, or "less small", than the former one)

1.8 x 10-2 = 0.0180. as you see. the "less small", or the biggest one of them all.

as you have negative exponent, it means you are dividing 1.8 by 100 (i.e. moving the dot towards left, two times).

if you had positive exponent for the scientific notation, power of tenths, you would be multiplying times 100 (moving the dot towards right) as below:

1.8 x (10)2 = 180

so. if you have negative exponent in scientific notation, move the dot toward left. if it is positive, toward right as many times the exponent tells you.

Answer from: Quest

step-by-step explanation: so we look at the formula un = a + (n-1)d

un is the amount of seats

n is the nth term aka row number

a is the original sequence/row number

d is the difference between each term

so in this a=22 d=6 and n can remain as n because we are simply formjng a rule

lets put these in the equation

un=22+(n-1)(6)

this is your answer to part a

however

lets double check that this is correct, we'll try putting it into the second term

un=a+(n-1)d

u2=22+(2-1)(6)

u2=22+6

u2=28

as the question said the second row has 28 seats so now we know the formula we made is correct.

as for part b

un=a+(n-1)

its asking us to find a row with 100 seats.

remember un was the amount of seata so all we do is substitute this in.

un=100 a=22 d=6

un=a+(n-1)d

100=22+(n-1)(6)

so now we simplify to find n which is the row number.

100-22=22+(n-1)(6)-22

78=(n-1)(6)

78/6=(n-1)(6)/6

13=n-1

13+1=n-1+1

14=n

so now we know that in the 14th row there is 100 seats.

hope this and that you understood! its my first time using the app or anyone out on here : p its also me to revise so a bunch! ask more questions if you dont understand ☺

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