Pre Calculus Hyperbola Question

Question

The sides of a nuclear power plant cooling tower form a hyperbola. The diameter of the bottom of the tower is 319 feet. The smallest diameter of the tower is 185 which is 380.5 feet above the ground. The tower is 501 feet tall.

Find the width of the tower at a height of 225 feet.

Answer 1

Since the sides of our nuclear power plant cooling tower form a hyperbola, we can deduce that we'll be working with a hyperbola with a horizontal transverse axis. Since the smallest diameter of our tower occurs 380.5 feet above the ground and the two points that correspond to the smallest diameter for a hyperbola with a horizontal transverse axis lie on the horizontal transverse axis, we can also deduce that the line y=380.5 is the horizontal transverse axis for our hyperbola. Since the smallest diameter of our tower is 185 feet and the smallest diameter for a hyperbola with a horizontal transverse axis is equal to 2*a, we can also deduce that 2*a = 185⇒a=92.5. Since there is insufficient information to determine the x-coordinate for the center of our hyperbola if we suppose that the x-coordinate for the center of our hyperbola is not equal to 0, we'll suppose that the x-coordinate for the center of our hyperbola is 0. The preceding information tells us that we'll need to work with the following hyperbola to solve this problem:

x^{2}/92.5^{2}-(y-380.5)^{2}/b^{2}=1

Since the diameter at the bottom of our tower is 319 feet and our hyperbola is symmetric about the y-axis, we know that the point (159.5,0) lies on our hyperbola. From this we can deduce that

159.5^2/92.5^2-(0-380.5)^2/b^{2}=1⇒159.5^2/92.5^2-(-380.5)^2/b^{2}=1

⇒159.5^2/92.5^2-(380.5)^2/b^{2}=1

⇒-(380.5)^2/b^{2}=1-159.5^2/92.5^2

⇒(380.5)^2/b^{2}=159.5^2/92.5^2-1

⇒(380.5)^2/[159.5^2/92.5^2-1]=b^{2}

⇒b^{2}=(380.5)^2/[159.5^2/92.5^2-1]

⇒b^{2}=73369.81841

⇒b=270.8686368,

which tells us that the hyperbola we'll need to solve our problem will simplify to

x^{2}/92.5^{2}-(y-380.5)^{2}/270.8686368^{2}=1.

If we now let y=225, we then find that

x^{2}/92.5^{2}-(225-380.5)^{2}/270.8686368^{2}=1

⇒x^{2}/92.5^{2}-(-155.5)^{2}/270.8686368^{2}=1

⇒x^{2}/92.5^{2}-(155.5)^{2}/270.8686368^{2}=1

⇒x^{2}/92.5^{2}=1+(155.5)^{2}/270.8686368^{2}

⇒x^{2}=92.5^2*[1+(155.5)^{2}/270.8686368^{2}]=11376.1052

⇒x=±106.6588262.

From this we can finally conclude that the width of our tower at a height of 225 feet is 2*106.6588262 feet = 213.3176524 feet ≈ 213.32 feet.