Windows 7 Join Domain
how would you do these problems?
1.) Find the domain and range of the function.
h(x)=sqrt4-x^2
2.) Find an expression for the function whose graph is the line segment joining the points (-7, 
and (8, -8).
3.) A spherical balloon with radius r inches has volume defined by the function below. Find a function that represents the amount of air required to inflate the balloon from a radius of r inches to a radius of r + 3 inches.
V(r)=4/3*pi*r^3
i got to 4/3*(3r^2+3r+27) and then i got stuck
lastly
4.) A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 22 ft, express the area A of the window as a function of the width x of the window.
i got to A = xy + 1/2 pi r^2
where
y = (22 - (pi +1) x) / 2
r = x/2
but what is A=
Ok,
1.) Remember, domain and range are looking at the values of x and y that are defined by (or in) the function. Probably the easiest way to answer this is to look at the graph. If you plug it into a graphing calculator, you will see the upper half of a circle. Otherwise you can plug in values for x until you notice that values smaller than -2 or greater than 2 yield negatives under the radical. Thus, the domain is [-2, 2]. Likewise, the related y-values will not go below 0 or above 2. Thus the range is [0, 2].
2.) For this one, I'm guessing you need the equation of the line formed by the two points. First find the slope between the points:
m = (-8 - 8)/(8 - (-7)) = -16/15
Now using this slope and one of the two points, plug into the point-slope form:
y - (-8) = -16/15*(x -
y + 8 = -16/15*x + 128/15
y = -16/15*x + 8/15
3.) Simply substitute in r + 3 for r. Thus you get:
V(r) = 4/3*pi*(r + 3)^3
= 4/3*pi*(r + 3)(r + 3)(r + 3)
= 4/3*pi*(r^2 + 6r + 9)(r + 3)
= 4/3*pi*(r^3 + 6r^2 + 9r + 3r^2 + 18r + 27)
= 4/3*pi*(r^3 + 3r^2 + 9r + 27) or,
= (4*pi*r^3)/3 + 4*pi*r^2 + 12*pi*r + 36*pi
I'm not sure what form you are looking for.
4.) You are close with your expression for y, but don't forget that it is a semicircle. So:
A = xy + 1/2*pi*r^2,
where y = (22 - (1/2*pi + 1)*x) / 2
and r = x/2
Now simply substitute your expressions for y and r into A. Thus you get:
A = x*[(22 - (1/2*pi + 1)*x)/2] + 1/2*pi*(x/2)^2
= 11x - (1/4*pi +1/2)*x^2 + 1/2*pi*1/4*x^2
= 11x - 1/4*pi*x^2 + 1/2*x^2 + 1/8*pi*x^2
= 11x - 1/8*pi*x^2 + 1/2*x^2
That should do it for you. Hope this helps.
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