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3 years ago

Write in point-slope form an equation of the line that passes through the point (−2,4) with slope −12.

Mathematics

1 answer:

alexgriva [62]3 years ago

7 0

Answer: y - 4 = -12 ( x + 2)

Step-by-step explanation:

The equation of a line in point - slope form is given as :

$y-y_{1}$ = m( $x-x_{1}$ )

Where m is the slope and ( $x_{1}$ , $y_{1}$ ) is the given point

substituting the values into the formula , we have:

y - 4 = -12 ( x - {-2})

y - 4 = -12 ( x + 2)

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3 years ago

At what point does the curve have maximum curvature? Y = 4ex (x, y) = what happens to the curvature as x → ∞? Κ(x) approaches as

MAXImum [283]

<u>Answer-</u>

At $x= \frac{1}{2304e^4-16e^2}$ the curve has maximum curvature.

<u>Solution-</u>

The formula for curvature =

$K(x)=\frac{{y}''}{(1+({y}')^2)^{\frac{3}{2}}}$

Here,

$y=4e^{x}$

Then,

${y}' = 4e^{x} \ and \ {y}''=4e^{x}$

Putting the values,

$K(x)=\frac{{4e^{x}}}{(1+(4e^{x})^2)^{\frac{3}{2}}} = \frac{{4e^{x}}}{(1+16e^{2x})^{\frac{3}{2}}}$

Now, in order to get the max curvature value, we have to calculate the first derivative of this function and then to get where its value is max, we have to equate it to 0.

${k}'(x) = \frac{(1+16e^{2x})^{\frac{3}{2} } (4e^{x})-(4e^{x})(\frac{3}{2}(1+e^{2x})^{\frac{1}{2}})(32e^{2x})}{(1+16e^{2x} )^{2}}$

Now, equating this to 0

$(1+16e^{2x})^{\frac{3}{2} } (4e^{x})-(4e^{x})(\frac{3}{2}(1+e^{2x})^{\frac{1}{2}})(32e^{2x}) =0$

$\Rightarrow (1+16e^{2x})^{\frac{3}{2}}-(\frac{3}{2}(1+e^{2x})^{\frac{1}{2}})(32e^{2x})$

$\Rightarrow (1+16e^{2x})^{\frac{3}{2}}=(\frac{3}{2}(1+e^{2x})^{\frac{1}{2}})(32e^{2x})$

$\Rightarrow (1+16e^{2x})^{\frac{1}{2}}=48e^{2x}$

$\Rightarrow (1+16e^{2x})}=48^2e^{2x}=2304e^{2x}$

$\Rightarrow 2304e^{2x}-16e^{2x}-1=0$

Solving this eq,

we get $x= \frac{1}{2304e^4-16e^2}$

∴ At $x= \frac{1}{2304e^4-16e^2}$ the curvature is maximum.

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3 years ago

In rectangle nice, the base is 2x and the height is x+1. the perimeter is 53 feet. find the height

mihalych1998 [28]

The perimeter equation for a rectangle is two times the base plus two times the height.
2B + 2H=P

Plug in the known values and solve for x.
2(2x) + 2(x+1)=53
4x+2x+2=53
6x=51
x=8.5

Now that we know what x is, put it back into the formula for the height of the rectangle to find the final answer.
H= x+1
H=8.5+1
H=9.5 ft

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