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3 years ago

A rock drops from a height of 60m. how long does it take for it to hit the ground?

Mathematics

1 answer:

laila [671]3 years ago

8 0

To solve for the time an object takes to hit the ground, the equation

y = y₀ + v₀t + ½gt², or just 0 = y₀ +½gt² since there is no initial velocity and the final y position is 0 (since it is on the ground). Therefore, t = (−2y₀/g)^½.

(-2*60/-9.8)^½ = 3.50 seconds.

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Calculus Problem

Roman55 [17]

The two parabolas intersect for

$8-x^2 = x^2 \implies 2x^2 = 8 \implies x^2 = 4 \implies x=\pm2$

and so the base of each solid is the set

$B = \left\{(x,y) \,:\, -2\le x\le2 \text{ and } x^2 \le y \le 8-x^2\right\}$

The side length of each cross section that coincides with B is equal to the vertical distance between the two parabolas, $|x^2-(8-x^2)| = 2|x^2-4|$ . But since -2 ≤ x ≤ 2, this reduces to $2(x^2-4)$ .

a. Square cross sections will contribute a volume of

$\left(2(x^2-4)\right)^2 \, \Delta x = 4(x^2-4)^2 \, \Delta x$

where ∆x is the thickness of the section. Then the volume would be

$\displaystyle \int_{-2}^2 4(x^2-4)^2 \, dx = 8 \int_0^2 (x^2-4)^2 \, dx \\\\ = 8 \int_0^2 (x^4-8x^2+16) \, dx \\\\ = 8 \left(\frac{2^5}5 - \frac{8\times2^3}3 + 16\times2\right) = \boxed{\frac{2048}{15}}$

where we take advantage of symmetry in the first line.

b. For a semicircle, the side length we found earlier corresponds to diameter. Each semicircular cross section will contribute a volume of

$\dfrac\pi8 \left(2(x^2-4)\right)^2 \, \Delta x = \dfrac\pi2 (x^2-4)^2 \, \Delta x$

We end up with the same integral as before except for the leading constant:

$\displaystyle \int_{-2}^2 \frac\pi2 (x^2-4)^2 \, dx = \pi \int_0^2 (x^2-4)^2 \, dx$

Using the result of part (a), the volume is

$\displaystyle \frac\pi8 \times 8 \int_0^2 (x^2-4)^2 \, dx = \boxed{\frac{256\pi}{15}}}$

c. An equilateral triangle with side length s has area √3/4 s², hence the volume of a given section is

$\dfrac{\sqrt3}4 \left(2(x^2-4)\right)^2 \, \Delta x = \sqrt3 (x^2-4)^2 \, \Delta x$

and using the result of part (a) again, the volume is

$\displaystyle \int_{-2}^2 \sqrt 3(x^2-4)^2 \, dx = \frac{\sqrt3}4 \times 8 \int_0^2 (x^2-4)^2 \, dx = \boxed{\frac{512}{5\sqrt3}}$

7 0

2 years ago

Goran made 6 identical necklaces, each having beads and a pendant. The total cost of the beads and pendants for all 6 necklaces

Tema [17]

  6b + 6p = 33.60 ; b=2.90

6(2.90) + 6p = 33.60

17.40 + 6p = 33.60
-17.40 -17.40

  6p = 16.20
/6 /6

  p = 2.70

ANSWER: each pendant costs $2.70

5 0

3 years ago

Read the following and answer the questions that follow.

user100 [1]

Answer:

Step-by-step explanation:

(7/12) of 840 attended the concert:

(7/12)*(840) = 490 students attended the Spring Concert.

(840 - 490) = 350 did not attend

(350/840) is the fraction of students who did not attend. This can be reduced to (35/84)

(350/840) = 0.4167 or 41.67% did not attend.

3 0

2 years ago

Maya has 150 caramel apples to sell. Each caramel apple is covered with one topping.

icang [17]

Answer:

Step-by-step explanation:

First you have to figure out how much each is worth by itself.

carmel =30

coco chips =50

coconut= 45

then add all of them you'd get 125

subtract 125 from 150 and get 25

4 0

3 years ago

Read 2 more answers

Simplify (-2) (-2) (-2) (-2)

kobusy [5.1K]

-2*4=-8
-8 is the correct answer

3 0

3 years ago

Read 2 more answers