Question

Cooling towers are used to remove or expel heat from a process. A cooling tower's walls are modeled by x squared over 289 minus quantity y minus 70 end quantity squared over 1600 equals 1 comma where the measurements are in meters. What is the width of the cooling tower at the base of the structure? round your answer to the nearest whole number.

Answer 1

To find the width of the cooling tower at the base, we need to solve the equation (y-70)^2/1600 = x^2/289 - 1 for x. The width of the cooling tower at the base is approximately 84 meters when rounded to the nearest whole number.

Step-by-step explanation:

To find the width of the cooling tower at the base, we need to solve the equation x^2/289 - (y-70)^2/1600 = 1 for x.

Rearranging the equation, we have (y-70)^2/1600 = x^2/289 - 1.

Since the equation is in the form of a hyperbola, we can use the properties of hyperbolas to find the width.

The formula for the width of the hyperbola is given by the equation 2a, where a is the distance from the center to a vertex.

From the equation (y-70)^2/1600 = x^2/289 - 1, we can see that the center of the hyperbola is at (0, 70) and the value of a is sqrt(289 + 1600).

Therefore, the width of the cooling tower at the base is approximately 2 * sqrt(289 + 1600), which is approximately 84 meters when rounded to the nearest whole number.

Answer 2

The width of the cooling tower at the base of the structure is 34 meters.

Solving word problems using hyperbola equation.

The given word problem can be represented using the equation of an hyperbola:

`(x^2)/(289)-((y-70)^2)/(1600)=1`

The general formula of the hyperbola equation is:

`((x-h)^2)/(a^2)-((y-k)^2)/(b^2)=1`

Relating the general formula with the hyperbola equation given, we have:

`a^2=289 ; \\ \\ a =√(289) ; \\ \\ a= 17`

`b^2=1600 ; \\ \\ b=√(1600) ; \\ \\ b = 40`

The center (h,k) is h = 0, k = 70. Now, from the equation, the width of the cooling tower at the base of the structure represents twice the length of the horizontal axis (a) of the hyperbola. i.e.

Width = 2a

Width = 2 × 17

Width = 34 meters.