Question

The formula for eccentricity, e, of an orbit is given below, where a is the length of the semi-major axis and b is the length of the major axis.

Solve the formula for eccentricity for the length of the semi-major axis.



The correct equation for the length of the semi-major axis is equation .

When the length of the major axis is 4 inches and the eccentricity of the orbit is , the length of the semi-major axis is inches (round to the nearest hundredth).

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Answer 1

To find the semi-major axis of an ellipse given the major axis and eccentricity, use the rearranged formula a = f/e, where f is half the length of the major axis. For a given major axis of 4 inches and an eccentricity of 0.5, the semi-major axis would be 4 inches.

Step-by-step explanation:

The question appears to involve finding the length of the semi-major axis of an ellipse given its eccentricity and the length of the major axis. The eccentricity formula is usually given by e = f/a, where e is the eccentricity, f is the distance from the center to one of the foci, and a is the semi-major axis. However, to find a, we need to rearrange this formula.

Given that the major axis is twice the semi-major axis, if the major axis is b, then a = b/2. Using the provided eccentricity, we calculate the semi-major axis as a = f/e, where f is half the major axis's length. If b is 4 inches and e is 0.5, then f is 2 inches (since f is half of b), and hence a = 2/e, which gives us a semi-major axis of 4 inches when e is 0.5.

Answer 2

Part 1)
`a=(b)/(√(1-e^2))`

Part 2)
`a=4.62\ in`

Step-by-step explanation:

The correct question in the attached figure

Part 1)

we have the formula

`e=\sqrt{1-(b^2)/(a^2)}`

where

e the eccentricity

a is the length of the semi-major axis

b is the length of the major axis

solve for a

That means

Isolate the variable a

`e=\sqrt{1-(b^2)/(a^2)}`

squared both sides

`e^2=1-(b^2)/(a^2)`

subtract 1 both sides

`e^2-1=-(b^2)/(a^2)`

Multiply both sides by a^2

`(e^2-1)a^2=-b^2`

Divide both sides by (e^2-1)

`a^2=(-b^2)/(e^2-1)`

Rewrite

`a^2=(-b^2)/(-(1-e^2))`

`a^2=(b^2)/((1-e^2))`

square root both sides

`a=\sqrt{(b^2)/((1-e^2))}`

simplify

`a=(b)/(√(1-e^2))`

Part 2) we have

`b=4\ in\\e=1/2`

Solve for a

substitute in the formula of part 1)

`a=(4)/(√(1-(1/2)^2))=4.62\ in`