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3 years ago

What is the gradient of the blue line.

Mathematics

2 answers:

denpristay [2]3 years ago

6 0

Answer:

Step-by-step explanation:

the Gradient is (5,2)

svet-max [94.6K]3 years ago

5 0

Its (5,2)!! Hope this helps i hope yiu have a great day!

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

xxTIMURxx [149]

First, rewrite the equation so that y is a function of x :

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

(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 ≤ x ≤ 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

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

7 0

3 years ago

One number is 3 less than a second number. Twice the second number is 12 more than 3 times the first. Find the two numbers.

Verdich [7]

6 and 3, 3 and 6, or 2 and 6

7 0

3 years ago

2x 90 and 69x 57.................................................

Gnom [1K]

Answer:

NICE DRAWING

Step-by-step explanation:

180

3933

4 0

3 years ago

Read 2 more answers

I need help answering

irga5000 [103]

For this equation, all you need to do is plug in the numbers to the equation...
(3/2)6 - 3 + (5/3)3
Then you simplify...
(18/2) - 3 + (15/3)
Then divide...
9 - 3 +5
Then subtract and add...
9-3=6
6+5=11
Answer: 11

3 0

3 years ago

Read 2 more answers

7th grade math help me please :)

Yuliya22 [10]

Answer:

$288

Step-by-step explanation:

16+8=24

$24 per book

24 x 12 = 288

7 0

3 years ago