BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071
BuyFind

Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

Solutions

Chapter 1.2, Problem 102E
To determine

The distance from the earth to the sun.

Expert Solution

Answer to Problem 102E

The distance from the earth to the sun is 1.496×1011m_.

Explanation of Solution

Formula used:

The distance is calculated using the formula d=(GM4π2)13T23.

Where, d is the distance from the earth to the sun, G is the gravitational constant, M is the mass of the sun, T is the period of the planet’s orbit in sec.

Division of powers:

“The division of two powers with the same base a and different exponents m and n is given by, aman=amn”.

That is, while dividing two powers with same base, the exponents are subtracted and the base will remain the same.

Law used:

Laws of exponents:

“To raise a power to a new power, multiply the exponents”.

(am)n=amn

Calculation:

Given that, the mass of the sun is 1.99×1030kg and the gravitational constant is 6.67×1011Nm2/kg2 and the period of earth’s orbit is about 365.25days.

It is known that, 1day=86400sec.

Use the above unit conversion to convert 365.25days into sec as follows.

365.25days=365.25×1day=365.25×86400sec=31557600sec

Note that, the period of earth’s orbit is about 31557600sec.

Substitute 6.67×1011Nm2/kg2 for G, 1.99×1030kg for M, 31557600sec for T to calculate d as follows.

d=(6.67×1011Nm2/kg2×1.99×1030kg4π2)13(31557600sec)23m=(13.2733×1030114π2)13(99862.5484)m=(0.3362×1019)13(99862.5484)m

Now, move decimal 1 place to the right and subtract 1 from the power of 10 in the first bracket of the above equation to simplify the expression.

d=(3.362×1018)13(99862.5484)m=(1.498×106)(99862.5484)m=149594.0975×106m

Similarly, move decimal 5 places to the left, add 5 to the power of 10 to express the above quantity in proper scientific notation as follows.

d=1.495940975×106+5m=1.495940975×1011m

Round off the above quantity up to three decimal places.

d=1.496×1011m

Therefore, the distance from the earth to the sun is 1.496×1011m_.

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