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2 years ago

James saw that the elevator started on floor 6 and five seconds later was on floor -1. What was the rate of change?

Mathematics

1 answer:

Margarita [4]2 years ago

8 0

Answer:

7/5 floors per second

Step-by-step explanation:

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A torus is formed by rotating a circle of radius r about a line in the plane of the circle that is a distance R (> r) from th

jeyben [28]

Consider a circle with radius $r$ centered at some point $(R+r,0)$ on the $x$ -axis. This circle has equation

$(x-(R+r))^2+y^2=r^2$

Revolve the region bounded by this circle across the $y$ -axis to get a torus. Using the shell method, the volume of the resulting torus is

$\displaystyle2\pi\int_R^{R+2r}2xy\,\mathrm dx$

where $2y=\sqrt{r^2-(x-(R+r))^2}-(-\sqrt{r^2-(x-(R+r))^2})=2\sqrt{r^2-(x-(R+r))^2}$ .

So the volume is

$\displaystyle4\pi\int_R^{R+2r}x\sqrt{r^2-(x-(R+r))^2}\,\mathrm dx$

Substitute

$x-(R+r)=r\sin t\implies\mathrm dx=r\cos t\,\mathrm dt$

and the integral becomes

$\displaystyle4\pi r^2\int_{-\pi/2}^{\pi/2}(R+r+r\sin t)\cos^2t\,\mathrm dt$

Notice that $\sin t\cos^2t$ is an odd function, so the integral over $\left[-\frac\pi2,\frac\pi2\right]$ is 0. This leaves us with

$\displaystyle4\pi r^2(R+r)\int_{-\pi/2}^{\pi/2}\cos^2t\,\mathrm dt$

Write

$\cos^2t=\dfrac{1+\cos(2t)}2$

so the volume is

$\displaystyle2\pi r^2(R+r)\int_{-\pi/2}^{\pi/2}(1+\cos(2t))\,\mathrm dt=\boxed{2\pi^2r^2(R+r)}$

6 0

3 years ago

Which choice is the explicit formula for the following geometric sequence? 0.2, -0.06, 0.018, -0.0054, 0.00162

Crazy boy [7]

Answer:

aₙ = 0.2(-0.3)⁽ⁿ⁻¹⁾

Step-by-step explanation:

The geometric sequence with the first term a, a common ratio r has the nth term given as

Tₙ = arⁿ⁻¹

where Tₙ is the nth term

From the given sequence

a = 0.2

r = -0.06/0.2

= -0.3

Hence the nth term

= 0.2 * -0.3ⁿ⁻¹

The right option is E

8 0

3 years ago

Bill paid $2.49 for 5 pounds of rice. What could be cost of 9 pounds

Nadya [2.5K]

Answer:

$\boxed{\sf Cost \ of \ 9 \ pounds \ rice =\$ 4.482}$

Given:

5 pounds rice cost $2.49

To Find:

Cost of 9 pounds rice

Step-by-step explanation:

5 pounds rice cost $2.49

1 pound rice will cost $ $\sf \frac{2.49}{5}$

So,

9 pounds rice cost = 1 pound rice cost × 9

= $ $\sf \frac{2.49}{5} \times 9$

= $0.498 × 9

= $4.482

3 0

3 years ago

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The math department is putting together an order for new calculators. The

alexgriva [62]

Answer:

Step-by-step explanation:

5 0

2 years ago

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If the bottom of the ladder is 3ft from a wall and the top of the ladder is 4ft from the ground, then how long is the ladder?

statuscvo [17]

Answer:

5ft

Step-by-step explanation:

$4^{2} +3^{2} =c^{2} \\16+9=25\\\sqrt{25} =5$

6 0

3 years ago