1- Find the coordinates of the point c, halfway between the points A(5,1) and B(-2,7) 2- What is the equation of the ellipse with foci at (0,-4) and (0,4) and the sum of its focal radii being 10?...

1- Find the coordinates of the point c, halfway between the points A(5,1) and B(-2,7)

2- What is the equation of the ellipse with foci at (0,-4) and (0,4) and the sum of its focal radii being 10?

3- The formula to find the total amount in your savings account (if interest is compounded continuously) after t years is A=Pe^rt.    a) how long does it take your money to double at 8% interest? (round to the nearest year)

b) at 5% interest? (round to the nearest year).

Thanks a lot

Asked on by eyehawk

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kingattaskus12's profile pic

kingattaskus12 | (Level 3) Adjunct Educator

Posted on

The answer on number 2 is this..

2.)  What is the equation of the ellipse with foci at `(0,-4)` and `(0,4)` and the sum of its focal radii being `10` ? 

The focal distance is `c=4` . The sum of the focal radii of the ellipse is constant and it's equal to the length of the major axis `2a ` according to the definition of the ellipse. This gives

`2a=10`

`a=5`

The equation relating the focal distance `c ` and lengths of semi major and minor axis `a` and `b` is

`a^2=b^2+c^2`

By plugging the acquired values, we have

`5^2 = b^2+4^2`

`25=b^2+16`

`b^2 = 25-16`

`b^2=9`

`b=3`

The coordinates of the foci (0,4) and (0,-4) shows that the ellipse has a vertical major axis with center at the origin. Thus, the equation of the ellipse in standard form is

`x^2/b^2+y^2/a^2=1`

`x^2/3^2+y^2/5^2=1`

`x^2/9+y^2/25=1`

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ishpiro's profile pic

ishpiro | College Teacher | (Level 1) Educator

Posted on

1) Since point C is hallfway between the points A(5, 1) and B(-2, 7), point C is the midpoint of the segment AB. According to the midpoint formula, the coordinates of the point C are

`x_c = (x_A + x_B)/2 = (5 + (-2))/2 = 1.5`

`y_c = (y_A + y_B)/2 = (1 + 7)/2 = 4`

` `The coordinates of point C are (1.5, 4).

3) The formula for continuous compound interest is `A = Pe^(rt)`

Here, A is the amount of money in the account, P is the principal (the amount originally put in), r is the interest rate (r = 8%, or 0.08, here) and t is the time.

If the amount doubles after time t, A = 2P. So the equation becomes

`2P = Pe^(0.08t)`

Canceling P, we get the exponential equation for time:

`e^(0.08t) = 2`

Taking natural log of both sides, get 0.08t = ln2

or `t = (ln2)/0.08 = 8.664...`

which rounds to 9 years. It takes 9 years for the money to double.

b) At 5% interest, the equation for time will be the same with 0.05 replacing 0.08:

`t = (ln2)/0.05 =13.862...` , which rounds to 14 years. 

It will take 14 years for the money to double.

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