Tutor profile: George K.

Write an equation for an ellipse centered at the point (7, -5), which has foci at (4, -5) and (10, -5), and vertices at (2, -5) and (12, -5).

$$ \frac {(x-7)^2}{25} +\frac {(y+5)^2}{16}=1 $$ The center determines the numbers that subtract and add x and y, respectively. The major axis of the ellipse is horizontal, as shown by the foci, and has radius 5, which is squared to 25. The minor axis is found by the difference of the squares of the major axis, 5 (25), and the foci, 3 (9).

A triangle has vertices A (-2, 4), B (6, 2), C (1, -1). Show that ABC is an isosceles right triangle.

In order to solve this, the slopes and lengths of each side of the triangle must be found. The slope of line segment AB is -1/4, the slope of line segment BC is 3/5, and the slope of line segment AC is -5/3. The slopes of AC and BC are negative reciprocals, therefore the sides make a right angle. The lengths are as follows: $$ AB = \sqrt68, BC = \sqrt34, and AC = \sqrt34 $$. Since BC = AC, the triangle is isosceles.

The expression $$ -9.8t^2 + 98t + 54 $$ represents the height, in meters, of a toy rocket t seconds after launch. Determine the maximum height of the rocket and when it occurs. Justify your response.

The maximum height of the rocket is 299 meters and it occurs when t = 5 seconds. I found this by determining the axis of symmetry using t = -b/2a, which gives the value t = 5. This represents the center of the quadratic expression, which is where the maximum (or minimum) occurs. By calculating the expression when t=5, the maximum was found to be 299. This can also be represented by the coordinate (5, 299), and by graphing the expression and finding the maximum point using a graphing calculator or other technology, but that's less impressive