Answer:
a = [tex]\frac{1}{2}[/tex]. Surface Area = [tex]\frac{4}{3}[/tex][tex]ft^{2}[/tex]. and area of the Dish = [tex]\frac{4}{3}[/tex][tex]ft^{2}[/tex]+pi[tex]4^{2}[/tex] = [tex]\frac{4}{3}[/tex][tex]ft^{2}[/tex]+50.27=51.6[tex]ft^{2}[/tex]
Step-by-step explanation:
(1) Constant. y(x) = a[tex]x^{2}[/tex] that is the curve that we need to rotate around the y axis to get the parabola with diameter of 8 feet and 2 meter depth that statement is translated in mathematics as x = -4 to 4 and y = 0 to 2.
y max = 2, x max = 4 setting up a equation with a unknown gives
2=a4 and a = [tex]\frac{1}{2}[/tex].
so we have now.
y(x) = [tex]\frac{1}{2}x^{2}[/tex] (Done with solving for a Constant).
(2) Surface Area.
Setting Up surface integral.
(i) range in x = 0 to 4.
(ii) range in y = 0 to 2.
integral is.
Integral(0-2)[{integral[(0-4)[tex]\frac{1}{32}[/tex][tex]x^{2}[/tex]]}]dy
Evaluating this integral gives. [tex]\frac{4}{3}[/tex][tex]ft^{2}[/tex].
and area is surface area + area of the circle with 8ft diameter.
= [tex]\frac{4}{3}[/tex][tex]ft^{2}[/tex]+pi[tex]4^{2}[/tex] = [tex]\frac{4}{3}[/tex][tex]ft^{2}[/tex]+50.27=51.6[tex]ft^{2}[/tex]...
Note the Difference between area and aurface area.!
