Exercise 32 Page 853

We know that one spherical helium-filled balloon with a diameter of 30 centimeters can lift a 14-gram object.

Since the diameter of the balloon is 30 centimeters, the radius of the sphere is r= 302=15 centimeters. Let's find out how much helium is needed to lift a 14-gram object. We will use the formula for the volume of a sphere.

V_\text{balloon}=\dfrac{4}{3}\pi r^3

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Substitute 15 for r and evaluate

Substitute

r= 15

V_\text{balloon}=\dfrac{4}{3}\pi ({\color{#0000FF}{15}})^3

CalcPowProd

Calculate power and product

V_\text{balloon}=\dfrac{4}{3}\cdot 3375\pi

MoveRightFacToNum

a/c* b = a* b/c

V_\text{balloon}=\dfrac{4\cdot 3375\pi}{3}

Multiply

V_\text{balloon}=\dfrac{13\,500\pi}{3}

CalcQuot

Calculate quotient

V_\text{balloon}=4500\pi

Therefore, we need 4500π cubic centimeters of helium to lift 14 grams. Let x be the amount of helium that we need to lift a person who weighs 65 kilograms, which is 65 000 grams. Let's write a proportion for the volume of helium and the mass that it can lift. rcl 65 000 grams & - & x cubic centimeters 14 grams & - & 4500π cubic centimeters Now let's write an equation, cross multiply, and solve it for x!

65 000/14=x/4500π

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Solve for x

CrossMult

Cross multiply

65 000* 4500π = 14x

Multiply

292 500 000π=14x

RearrangeEqn

Rearrange equation

14x=292 500 000π

DivEqn

.LHS /14.=.RHS /14.

x=292 500 000π/7

UseCalc

Use a calculator

x=65 636 846.512...

RoundDec

Round to 2 decimal place(s)

x≈ 65 636 846.51

Therefore, we need about 65 636 846.51 cubic centimeters of helium to lift a 65-kilogram person. We are asked to find the size of a balloon that can lift a 65-kilogram person, which will have a volume about 65 636 846.51 cubic centimeters. Let r be its radius.