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
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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
The equation relating the focal distance `c ` and lengths of semi major and minor axis `a` and `b` is
By plugging the acquired values, we have
`5^2 = b^2+4^2`
`b^2 = 25-16`
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
Thank you veryyyyyyy much. This make more sense to me. thanks again.
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.
Thanks a lot
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