Iam so stuck on this problem. me.

  20.  170 cm³

  21.  The ice cream sandwich (rectangular prism) has the greatest volume.

Step-by-step explanation:

20. You are asked to find the difference between the volumes of three spheres and the volume of the cylinder containing them. It is appropriate to make use of the formulas for the volume of a sphere and the volume of a cylinder.

Sphere

The volume is given by ...

  V = (4/3)πr³

You have 3 spheres, each with a radius of 3 cm, so their total volume is ...

  3V = 3·(4/3)·π·(3 cm)³ = 108π cm³

Cylinder

The volume is given by ...

  V = πr²·h

You have a radius of 3 cm and a height of 18 cm, so the volume of the cylinder is ...

  V = π·(3 cm)²·(18 cm) = 162π cm³

Empty Space

The volume of the space in the cylinder not occupied by the balls will be ...

  space = (162π cm³) - (108π cm³) = 54π cm³ ≈ 169.64 cm³ ≈ 170 cm³

The space not occupied by balls is 170 cm³.

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21. The formula for the volume of a cone with a certain radius and height is ...

  V = (1/3)πr²·h

If we let r = d/2, where d is the diameter, then the volume is ...

  V = (1/3)π(d/2)²·h = π/12·d²·h ≈ 0.262·d²·h

The formula for the volume of a cylinder with a certain radius and height is ...

  V = πr²·h

Again, using r = d/2, the volume is ...

  V = π(d/2)²·h = (π/4)·d²·h ≈ 0.785·d²·h

The formula for the volume of a rectangular prism with a side length of d and a height of h is ...

  V = d²·h

That is, the volume of the cone (not counting the ice cream on top) is about 0.262 that of the prism, and the volume of the cylinder is about 0.785 that of the prism. Obviously, the prism shape contains the greatest volume.

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The conclusion doesn't change if we add the volume of a hemisphere of ice cream on top of the cone. That volume is ...

  hemisphere volume = (2/3)πr³ = (2/3)π(d/2)³ = π/12·d³ = π/12·d²·d

Now, the volume of the cone and hemisphere together will be ...

  topped cone volume = π/12·d²·h + π/12·d²·d = (π/12)·d²·(h+d)

That is, we have multiplied our previous assessment of the volume of the ice cream cone by (h+d)/h = (6+4)/6 = 5/3. This makes the multiplier of d²·h be about π/12·(5/3) ≈ 0.436, still less than half the volume of the ice cream bar.

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Note that we have chosen to look at the relative sizes of the cone, cylinder and prism in a generic way. You can fill in the numbers and compare volumes in cubic inches. The conclusion will be the same. We hope this method helps develop some intuition about the relative sizes of cones, cylinders, and prisms that have the same dimensions.

step-by-step explanation:

30% of $420 is 0.30($420) = $126 and $420 + $126 = $546.

or, 130% of $420 is 1.3($420) = $546.

notice that this is not the $600 original list price with which we started.

why doesn't this work?

the method does not work because 30% of $420 is not the same amount as the 30% of $600. it may seem obvious when stated this way, but it is a very common error made on problems of this type. we cannot just use the same percent and get back to the original amount because we are taking that percent of a different value.

to work problems of this type correctly, it is generally best to write an equation for the situation and then solve.

example: since (100% – 30%)(original price) = (sale price)

70% of the   (original price) is $420.

0.7x   = 420

equation image indicator

x = 600

the original cost was $600.

we may also solve this problem using the two-step method. working the problem in two steps requires us to use more algebra skills. we have to know that x = 1x (identity property of multiplicationtext annotation indicator) and that 1x – 0.3x = 0.7x (subtracting like terms). then we solve the problem as follows:

(original price) – 30%(original price) = sale price

x – 0.3x = 420

0.7x = $420

equation image indicator

x = 600

the original cost was $600.  

note that with both of the above methods we do obtain the correct original amount.

example: after a 5% pay raise, hermione is earning $22,680 per year. what was she earning before the pay raise?

105% of (original salary) = current salary

1.05x = 22,680

equation image indicator

x = 21,600

hermione earned $21,600 per year before the pay raise.

this is an example so maybe you will get how to at least do it!

-13

step-by-step explanation:

f(x)=2x-5 for f(-4)

f(-4)   means let x=-4

f(-4) = 2(-4) -5

f(-4) =-8-5

      = -13