Intermediate Algebra (11th edition)

A lithotripteris a machine used to crush kidney stones using shock waves. The patient is
placed in an elliptical tub with the kidney stone at one focus of the ellipse. A beam is projected
from the other focus to the tub, so that it reflects to hit the kidney stone. See the figure.

49.Suppose a lithotripter is based on the ellipse with equation

How far from the center of the ellipse must the kidney

stone and the source of the beam be placed? (Hint: Use the

fact that since here.)

50.Rework Exercise 49if the equation of the ellipse is

(Hint:Write the equation in fractional form by dividing

each term by 36, and use since here.)

Solve each problem.

51.An arch has the shape of half an ellipse. The equation of the
ellipse is

where xand yare in meters.

(a)How high is the center of the arch?

(b) How wide is the arch across the bottom?

52.A one-way street passes under an overpass, which is in the
form of the top half of an ellipse, as shown in the figure.
Suppose that a truck 12 ft wide passes directly under the
overpass. What is the maximum possible height of this
truck?

100 x^2 + 324 y^2 =32,400,

c^2 =b^2 - a^2 , b 7 a

9 x^2 + 4 y^2 =36.

c^2 =a^2 - b^2 , a 7 b

x^2

36

+

y^2

9

=1.

SECTION 11.2 The Circle and the Ellipse 649

Kidney

stone

Focusing Emitter

reflector

Shock

waves

The top of an ellipse is illustrated in

this depiction of how a lithotripter

crushes a kidney stone.

y

x

NOT TO SCALE

2 20 ft 0 ft

1 15 ft 5 ft

NOT TO SCALE

53.The orbit of Mars is an ellipse with the
sun at one focus. For xand yin millions
of miles, the equation of the orbit is

(Source:Kaler, James B., Astronomy!,

Addison-Wesley.)

(a)Find the greatest distance (the

apogee) from Mars to the sun.

(b)Find the least distance (the perigee)

from Mars to the sun.

54.The orbit of Venus around the sun (one

of the foci) is an ellipse with equation

where xand yare measured in millions

of miles. (Source: Kaler, James B.,

Astronomy!, Addison-Wesley.)

(a)Find the greatest distance between

Venus and the sun.

(b)Find the least distance between

Venus and the sun.

x^2

5013

+

y^2

4970

=1,

x^2

141.7^2

+

y^2

141.1^2

=1.

In Exercises 53 and 54, see FIGURE 13and use the fact that c^2 =a^2 - b^2 ,where .a^27 b^2

a – c

Perigee

Apogee

a + c

Mars

NOT TO SCALE

Orbit

Center

of

ellipse

c^2 = a^2 – b^2

Sun