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
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.
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