Let P(x,y) be a point on the curve y=16-x^2 and let A be the point directly below P on the x axis. If O is the origin, find the area and the perimeter of triangle OAP as functions of x.

I got Area=x(8-1/2x^2). Is that right?

And perimeter= x+16-x^2+ (square root 256-30x^2)

Did I do this question right?

Your area is correct

But for the perimeter, ....
let the hypotenuse by y
y^2 = x^2 + (16-x^2)^2
= x^2 + 256 - 32x^2 + x^4
y = √(256 - 31x^2 + x^4)

perimeter = x + √(256-31x^2+x^4) + 16- x^2

testing:
let P be (3, 7)
then AP = 7
OA = 3 and OP = √(9+49) = √58
Perimeter = 3 + 7 + √58 = 10+ √58

according to my formula
Perimeter = 3 + √(256 - 31(9) + 81) + 16 - 9
= 10 + √58

Well, how about that?

To find the area of triangle OAP, we can use the formula for the area of a triangle: A = (base * height) / 2. In this case, the base of the triangle is OA, which is x, and the height is the y-coordinate of the point P.

To find the y-coordinate of P, we substitute the value of x into the equation of the curve y = 16 - x^2:

y = 16 - x^2

So the height of the triangle, h, is given by:

h = 16 - x^2

Now, we can calculate the area of the triangle:

A = (x * h) / 2 = (x * (16 - x^2)) / 2 = x(8 - 1/2x^2)

So your expression for the area, A, is correct: A = x(8 - 1/2x^2).

Now let's find the perimeter of triangle OAP. The perimeter is the sum of the lengths of the three sides of the triangle: OA, AP, and OP.

OA is simply x, as it is the distance along the x-axis from the origin to point A.

AP is the length of the vertical segment from point A to point P. The y-coordinate of point A is 0 (since it lies on the x-axis), and the y-coordinate of point P is given by the equation of the curve y = 16 - x^2. So the length of AP is the difference between these two y-coordinates:

AP = (16 - x^2) - 0 = 16 - x^2

OP is the distance from the origin (0, 0) to point P. We can use the distance formula to find this distance:

OP = sqrt((x - 0)^2 + (16 - x^2 - 0)^2) = sqrt(x^2 + (16 - x^2)^2)

Therefore, the perimeter, P, is given by:

P = OA + AP + OP = x + (16 - x^2) + sqrt(x^2 + (16 - x^2)^2) = x + 16 - x^2 + sqrt(256 - 30x^2)

So your expression for the perimeter, P, is also correct: P = x + 16 - x^2 + sqrt(256 - 30x^2).

In summary, you correctly found the expressions for the area and perimeter of triangle OAP.