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2 years ago

Who wrote down the equation for the law of gravitation?

Mathematics

1 answer:

aalyn [17]2 years ago

7 0

It was the English Scientist Sir Isaac Newton.

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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]

Answer-

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

Solution-

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.

6 0

2 years ago

What is 28.2% of 1128

kap26 [50]

28.2% of 1128
(28.2%)(1128)
(28.2/100)(1128)
(0.282)(1128)
318.096

The answer is 318.096

4 0

2 years ago

easy car rental charges $40 a day plus $.25 per mile race. rent a car charges $25 a day plus $.45 per mile. what number of miles

DiKsa [7]

A: cost= 3*55+.35m
B: cost=3*50+.40m

set the costs equal, and solve for m.

6 0

2 years ago

Alexis has $22 75 to spend at the movies she buys a ticket for $8.500 a popcorn for $725 and a candy bar for $5 how much money d

vaieri [72.5K]

Answer:

$2.00

Step-by-step explanation:

Alexis spends:

$8.50 + $7.25 + $5 = $20.75

Money left:

$22.75 - $20.75 = $2.00

6 0

2 years ago

Read 2 more answers

What is the C of a circle that has a radius of 4.5

Jlenok [28]

Step-by-step explanation:

$C = 2\pi \: r \\ = 2 \times 3.14 \times 4.5 \\ = 3.14 \times 9 \\ =28.26 \: units$

5 0

2 years ago