Answer:
[tex]91.67 ft^3[/tex]
Step-by-step explanation:
We are given that radius of hemispherical dome=50 feet
Number of coats of paint=7
Thickness of each coat of paint=[tex]\frac{1}{100} in[/tex]
We have to find the volume of paint by using linear approximation
Volume of hemisphere=[tex]\frac{2}{3}\pi r^3[/tex]
Differentiate w.r.t r
Then, we get [tex]\frac{dv}{dr}=2\pi r^2[/tex]
[tex]\delta r=\frac{1}{100\times 12}\times 7=\frac{7}{1200} feet [/tex]
By approximation
[tex]\delta V=\frac{dV}{dr}\delta r[/tex]
Substitute the values then we get
[tex]\delta V=2\pi r^2\cdot \frac{7}{1200}[/tex]
[tex]\delta V=2\times \frac{22}{7}\times (50)^2\times \frac{7}{1200}=91.67 ft^3[/tex]
The volume of paint needed for the job using linear approximation is; δV = 91.67 ft³
We are given;
Radius of hemispherical dome; r = 50 feet
Number of coats of paint; n = 7
Thickness of each coat of paint; t = 1/100 inch = 1/1200 ft
Volume of hemisphere is;
V = ²/₃πr³
Differentiating with respect to r gives;
dV/dr = 2πr²
δr = 1/1200 ft * 7
δr = 7/1200 ft
By linear approximation;
δV = dv/dr * δr
δV = 2π(50)² * 7/1200
δV = 91.67 ft³
Read more about hemisphere volume at; https://brainly.com/question/23979348
