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3 years ago

Each lap around the track at emma's high school is 1/4 mile. How many laps will emma need to jog if she wants to jog 3 miles?

Mathematics

1 answer:

Luda [366]3 years ago

3 0

Answer:

Step-by-step explanation:

equation:

3/1 divided by 1/4= 3/1 x 1/4= 3/1 x 4/1= 12

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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

In Whitehorse, the high temperature for the day was 7°C, and in Yellowknife the high for the same day was -8°C. What was the dif

Lesechka [4]

The difference between the two cities are that each temp is different their 14c away because their is a minus on the eight

6 0

3 years ago

Graph the linear equation by any method. 2x=-4y+2 help me graph this

almond37 [142]

First give 0 value to x and find y, then give 0 value to y and find x. the values you find will be the point that is on the line. then for example give 7 to x and find y. this will be the second point and draw the line. you just have to do the first step and try the second one

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3 years ago

3 Caleb claims that when you multiply a number by 10, you can just write a 0 at the right end of the number. Caleb gives the exa

devlian [24]

Answer:

They are both correct

Step-by-step explanation:

It's interpretation of the concept. In common practice multiplying by 10 means you add a zero and that's what we teach 10 year olds. But what does adding a zero mean? It means adding a place value and that's what you learn when you are

So in both examples just add a decimal:

2.0 x 10 = 20.

0.2 x 10 = 2.0

In conclusion, it depends on the level of precision required in the discussion and the level of maturity as to who is correct in their examples. If you are teaching 3rd graders math, by all means you add a zero. If you are discussing physics you are shifting over a place value.

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3 years ago

Meagan is taking piano and gymnastics lessons. She has a piano lesson every 13 days and a gymnastics lesson every 6 days. When w

liraira [26]

the answer is 2

there are 13 days, 6x2=12. so the answer is 2.

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3 years ago