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castortr0y [4]
3 years ago
13

1.) Find the length of the arc of the graph x^4 = y^6 from x = 1 to x = 8.

Mathematics
1 answer:
xxTIMURxx [149]3 years ago
7 0

First, rewrite the equation so that <em>y</em> is a function of <em>x</em> :

x^4 = y^6 \implies \left(x^4\right)^{1/6} = \left(y^6\right)^{1/6} \implies x^{4/6} = y^{6/6} \implies y = x^{2/3}

(If you were to plot the actual curve, you would have both y=x^{2/3} and y=-x^{2/3}, but one curve is a reflection of the other, so the arc length for 1 ≤ <em>x</em> ≤ 8 would be the same on both curves. It doesn't matter which "half-curve" you choose to work with.)

The arc length is then given by the definite integral,

\displaystyle \int_1^8 \sqrt{1 + \left(\frac{\mathrm dy}{\mathrm dx}\right)^2}\,\mathrm dx

We have

y = x^{2/3} \implies \dfrac{\mathrm dy}{\mathrm dx} = \dfrac23x^{-1/3} \implies \left(\dfrac{\mathrm dy}{\mathrm dx}\right)^2 = \dfrac49x^{-2/3}

Then in the integral,

\displaystyle \int_1^8 \sqrt{1 + \frac49x^{-2/3}}\,\mathrm dx = \int_1^8 \sqrt{\frac49x^{-2/3}}\sqrt{\frac94x^{2/3}+1}\,\mathrm dx = \int_1^8 \frac23x^{-1/3} \sqrt{\frac94x^{2/3}+1}\,\mathrm dx

Substitute

u = \dfrac94x^{2/3}+1 \text{ and } \mathrm du = \dfrac{18}{12}x^{-1/3}\,\mathrm dx = \dfrac32x^{-1/3}\,\mathrm dx

This transforms the integral to

\displaystyle \frac49 \int_{13/4}^{10} \sqrt{u}\,\mathrm du

and computing it is trivial:

\displaystyle \frac49 \int_{13/4}^{10} u^{1/2} \,\mathrm du = \frac49\cdot\frac23 u^{3/2}\bigg|_{13/4}^{10} = \frac8{27} \left(10^{3/2} - \left(\frac{13}4\right)^{3/2}\right)

We can simplify this further to

\displaystyle \frac8{27} \left(10\sqrt{10} - \frac{13\sqrt{13}}8\right) = \boxed{\frac{80\sqrt{10}-13\sqrt{13}}{27}}

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The product of three and a number decreased by ten is thirteen
monitta

Answer:

3x - 10 = 13 would be the verbal expression where x = a number

Hope this helps!

5 0
2 years ago
5g + 7g and g(5 + 7) when g=6
oksian1 [2.3K]

Answer:

Value of equation = 144

Step-by-step explanation:

Given:

5g + 7g + g(5 + 7)

when g = 6

Find:

Value of equation

Computation:

5g + 7g + g(5 + 7)

5g + 7g + g(12)

5(6) + 7(6) + (6)(12)

30 + 42 + 72

Value of equation = 144

4 0
3 years ago
Find the value of given expression<br><br><img src="https://tex.z-dn.net/?f=%20%5Csqrt%7B655%20%20%5Ctimes%20655%7D%20" id="TexF
S_A_V [24]

Answer:

\sqrt{655*655} =655

Step-by-step explanation:

655*655 is pretty much equal to 655²

the square root of a number to the power of 2 is equal to the number without the power

so √655² would be equal to 655

4 0
2 years ago
Read 2 more answers
true or false:the points (6,13),(21,33),(99,137)all lie on the-same line. the equation of the line is y=4/3x +5
Anit [1.1K]

The answer is true.

Step-by-step explanation:

To find the points all lie on the same line, we need to substitute the points in the equation of the line, to determine if the values on both sides of the equation are equal.

Substituting the point (6,13) in the equation of the line, we get,

\begin{aligned}y &=\frac{4}{3} x+5 \\13 &=\frac{4}{3}(6)+5 \\&=4(2)+5 \\&=8+5 \\13 &=13\end{aligned}

Thus, the values on both sides are equal. The point (6,13) lie on the same line.

Substituting the point (21,33) in the equation of the line, we get,

\begin{aligned}y &=\frac{4}{3} x+5 \\33 &=\frac{4}{3}(21)+5 \\&=4(7)+5 \\&=28+5 \\33 &=33\end{aligned}

Thus, the values on both sides are equal. The point (21,33) lie on the same line.

Substituting the point (99,137) in the equation of the line, we get,

\begin{aligned}y &=\frac{4}{3} x+5 \\137 &=\frac{4}{3}(99)+5 \\&=4(33)+5 \\&=132+5 \\137 &=137\end{aligned}

Thus, the values on both sides are equal. The point (99,137) lie on the same line.

Thus, all the three points lie on the same plane.

Hence, the answer is true.

6 0
3 years ago
Round 82.265 to the nearest whole number. Do not write extra zeros.<br><br>​
patriot [66]

Answer:

82

Step-by-step explanation:

8 0
3 years ago
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