MATHEMATICS HIGH SCHOOL

A cylindrical water bottle with a height of 20 cm and diameter of 5.5 cm is filled with water until it is full. Vincent wants to transfer the water in the bottle into a cubic container. State the minimum length of a side of the cube. Can anyone help to explain this question for me?

Answers

Answer 1

Answer:

Step-by-step explanation:

the volume of a cylinder is

ground area × height

the ground area is a circle, and the area of a circle is

pi×r²

with r being the radius, which is always half of the diameter.

so, in our case

r = 5.5/2 = 2.75 cm

height = 20 cm

so, the volume of the cylinder is

pi×2.75² × 20 = pi×7.5625×20 = 475.1658889... cm³

the volume of a cube is

(side length)³

and the side length of a cube is therefore the cubic root of the volume.

now the volume of the cube has to be at least a large as the volume of the cylinder to be able to contain all the water from the cylinder.

so, the side length has to be at least the cubic root of that volume (475.1658889... cm³).

which is 7.803361957... cm.

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Ex 2.1120) A curve y''=12x-24 and a stationary point at (1,4). evaluate y when x=2.

If the half-life of a unstable isotope is 10,00 years and only 1/8 of the radioactive parent remains how old is the sample? A.10,000 B. 20,000 C. 30,000 D. 40,000

Answers

Answer:

C. 30,000

Step-by-step explanation:

We know that, the exponential function for decay is,

$y=ae^(rt)$

where,

y = the amount after time t,

a = initial amount,

r = rate of decay.

The half-life of a unstable isotope is 10,000 years, so

$\Rightarrow (1)/(2)=1\cdot e^(r\cdot 10000)$

$\Rightarrow (1)/(2)=e^(r\cdot 10000)$

$\Rightarrow \ln (1)/(2)=\ln e^(r\cdot 10000)$

$\Rightarrow -\ln 2={r\cdot 10000}\cdot \ln e$

$\Rightarrow -\ln 2={r\cdot 10000}\cdot 1$

$\Rightarrow r=(-\ln 2)/(10000)$

Now the function becomes,

$y=ae^{(-\ln 2)/(10000)\cdot t}$

Now, only 1/8 of the radioactive parent remains, so

$\Rightarrow (1)/(8)=1\cdot e^{(-\ln 2)/(10000)\cdot t}$

$\Rightarrow (1)/(8)=e^{(-\ln 2)/(10000)\cdot t}$

$\Rightarrow \ln (1)/(8)=\ln e^{(-\ln 2)/(10000)\cdot t}$

$\Rightarrow -\ln 8={(-\ln 2)/(10000)\cdot t}\cdot \ln e$

$\Rightarrow -\ln 8={(-\ln 2)/(10000)\cdot t}$

$\Rightarrow t=(-\ln 8\cdot 10000)/(-\ln 2)$

$\Rightarrow t=30,000$

IF the half-life of an unstable isotope is 10,000 years and only 1/8 of the radioactive parent remains, the sample is 30,000 years old.

Find the geometric mean between 4 square root of 3 and 10 square root of 3

Answers

Hello,
4√3*10√3=40*3=120

AB= 12m, BC= 16 cm and AD= 13 m.
Find the area of the shaded region.

Answers

Answer:

100π - (96 + (13/2)√231 ) m^2

Step-by-step explanation:

We will solve this by finding the area of the triangles (the quadrilateral is cut by the diameter) and subtract it from the area of the circle.

There is a circle theorem which states that if a triangle is in the semicircle and the hypothenuse extends to the length of the diameter, the angle at B here is 90 degrees.

Since angle B is 90 degrees we can use the Pythagorean theorem (a^2 + b^2 = c^2) to find Line AC.

so 144 + 256 = 400

√400 = 20

So diameter is 20

Finding the area of the top triangle:

1/2(12x16) = 96cm^2

For the bottom triangle, we need the side DC to find the area. To find it we will apply the theorem once again.

400 = 169 + DC^2

DC^2 = 231

DC = √231

so the area of the bottom triangle is 1/2(13√231) = (13/2)√231

Now we add the area of the triangles for the quadrilateral and subtract from the circle area.

The circle area is 100π (πr^2)

Hence, the area of the shaded region is:

100π - (96 + (13/2)√231 ) m^2

Calculate the radius of the sphere with the given volume.

Volume=4/81π ft^3

Answers

Answer:

(∛3)/2 ft = r

Step-by-step explanation:

Start with the formula for the volume of a sphere: V = (4/3)πr³. Solve this for r³ and then take the cube root of the result:

V = (4/3)πr³ => (3/4)V = πr³ => (------- = r³

4π

3 · 4/81π ft^3 3 · 4π ft³ 3 ft³

Here we have r³ = -------------------- = ---------------- = -------------

4π 8(4π) 8

and so the radius is r = ∛[ (3/8) ft³] = (∛3)/2 ft = r

T/2+t/3=5 solve for t

Answers

$(t)/(2)+(t)/(3)=5|\cdot6\n 6\cdot(t)/(2)+6\cdot(t)/(3)=6\cdot5\n3t+2t=30\n 5t=30|/5\n t=(30)/(5)\nt=6$

Designer Dolls, Inc. found that the number N of dolls sold varies directly with their advertising budget A and inversely with the price P of each doll. The company sold 5200 dolls when $26,000 was spent on advertising and the price of a doll was set at $30. Determine the number of dolls sold when the amount spent on advertising is increased to $52,000. Round to the nearest whole number.

Answers

Answer:

The number of dolls sold 10,400

Step-by-step explanation:

Given: The number N of dolls sold varies directly with their advertising budget A and inversely with the price P of each doll.

N = k(A/P), where k is the constant.

Now we have to find k.

Given: N = 5200, A = 26,000 and P = 30

5200 = k (26000/30)

5200 = k(866.67)

k = 5.99, when we round off we get k = 6

Now let's find the number of dolls sold when the ad amount increase to $52,000

Now plug k = 6, A = 52000 and p = 30

N = 6(52000/30)

N = (52000/5)

N = 10.400

Therefore, the number of dolls sold 10,400

Hope this will helpful.

Thank you.

I think 104,000 is the answer because when 26,000 is double it's 52,000 so I doubled 52,000 and 104,000 is what I got.

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