Which one is smaller than 2500ml and 25ml

Find the equation of the tangent line to the curve (a lemniscate)

olya-2409 [2.1K]

Answer:

$m=\frac{9}{13}$ and $b=\frac{40}{13}$

Step-by-step explanation:

The equation of curve is

$2(x^2+y^2)^2=25(x^2-y^2)$

We need to find the equation of the tangent line to the curve at the point (-3, 1).

Differentiate with respect to x.

$2[2(x^2+y^2)\frac{d}{dx}(x^2+y^2)]=25(2x-2y\frac{dy}{dx})$

$4(x^2+y^2)(2x+2y\frac{dy}{dx})=25(2x-2y\frac{dy}{dx})$

The point of tangency is (-3,1). It means the slope of tangent is $\frac{dy}{dx}_{(-3,1)}$ .

Substitute x=-3 and y=1 in the above equation.

$4((-3)^2+(1)^2)(2(-3)+2(1)\frac{dy}{dx})=25(2(-3)-2(1)\frac{dy}{dx})$

$40(-6+2\frac{dy}{dx})=25(-6-2\frac{dy}{dx})$

$-240+80\frac{dy}{dx})=-150-50\frac{dy}{dx}$

$80\frac{dy}{dx}+50\frac{dy}{dx}=-150+240$

$130\frac{dy}{dx}=90$

Divide both sides by 130.

$\frac{dy}{dx}=\frac{9}{13}$

If a line passes through a points $(x_1,y_1)$ with slope m, then the point slope form of the line is

$y-y_1=m(x-x_1)$

The slope of tangent line is $\frac{9}{13}$ and it passes through the point (-3,1). So, the equation of tangent is

$y-1=\frac{9}{13}(x-(-3))$

$y-1=\frac{9}{13}(x)+\frac{27}{13}$

Add 1 on both sides.

$y=\frac{9}{13}(x)+\frac{27}{13}+1$

$y=\frac{9}{13}(x)+\frac{40}{13}$

Therefore, $m=\frac{9}{13}$ and $b=\frac{40}{13}$ .

5 0

3 years ago

I do not understand this question, what do i do??

DiKsa [7]

Answer:

90 shoppers have cookies or fruits as their favorite snack.

Step-by-step explanation:

First add 15% and 15% to get a total of 30% then multiply 300 by 30 to get 9000 and divide that by 100 to get your answer of 90.

8 0

3 years ago

Read 2 more answers

The maximum afternoon temperature in Granderson is modeled by t=60-30 cos (x<img src="https://tex.z-dn.net/?f=%20%5Cpi%20" id="T

kicyunya [14]

The equation is $t=60-30 cos (x \frac{\pi}{6})$

the -30 expression, is subtracting, if negative, from the 60
if the cosine returned is negative, you'd end up with a +30,
negative * negative = positive

and then the 30 expression will ADD to the 60 amount
so the temperature "t" is highest, when the 30 expression, is
positive and and it's highest

when does that happen, when cosine is negative and at its highest,
well, cosine range is $-1\le cos(\theta ) \le 1$
so the lowest value cosine can provide is -1, when is cosine -1?
well, at $\pi$

so... let's find a value that makes that expression to $cos(\pi)$

$\bf t=60-30 cos (x \frac{\pi}{6}) \qquad x=6
\\\\
thus
\\\\
t=60-30 cos (6 \frac{\pi}{6})\implies t=60-30 cos (\pi )
\\\\
t=60-30[-1]\implies t=60+30\implies t=90$

----------------------------------------------------------------------------------------
so...for August, that'll mean

$\bf t=60-30 cos (x \frac{\pi}{6}) \qquad x=7
\\\\
t=60-30 cos (\frac{7\pi}{6})\implies t=60-30\left( -\cfrac{\sqrt{3}}{2} \right)
\\\\
t=60+(15\cdot \sqrt{3})\implies t\approx85.98^o$

4 0

3 years ago

Travis has a customer who wants to purchase a portable CD player for $45 at a 20% discount. What is the amount of the discount?

kherson [118]

Just do 45x0.2 then take that number off of $45 thats your answer

8 0

3 years ago

Read 2 more answers

The prism shown has a volume of 798 cm3.

a_sh-v [17]

Answer:

10.5 cm

Step-by-step explanation:

Volume = base area × height

798 = (9.5 × 8) × height

height = 798/76

height = 10.5

4 0

3 years ago