Find an equation that models a hyperbolic lens with a=16 inches and foci that are 40 inches apart. Assume that the center of the hyperbola is the origin and the transverse axis is vertical
But did you get it now
To find an equation that models a hyperbolic lens, we need to consider the properties of a hyperbola and use the given information.
A hyperbola with a vertical transverse axis and centered at the origin has the equation in its standard form:
((y^2)/(a^2)) - ((x^2)/(b^2)) = 1,
where "a" and "b" are positive constants related to the shape and dimensions of the hyperbola.
In this case, we are given that "a" is 16 inches, so we can substitute that into the equation:
((y^2)/(16^2)) - ((x^2)/(b^2)) = 1.
To determine the value of "b," we need to use the given information about the foci. The distance between the foci is given as 40 inches.
For hyperbolas, the relationship between the distance from the center to the foci ("c") and the value of "a" is given by the formula:
c^2 = a^2 + b^2.
In this case, the distance between the foci is 40 inches, so we have:
40^2 = 16^2 + b^2.
Solving for "b^2" gives us:
1600 = 256 + b^2,
1344 = b^2,
b = sqrt(1344).
Now we have all the information required to write the equation of the hyperbola:
((y^2)/(16^2)) - ((x^2)/(sqrt(1344)^2)) = 1.
Simplifying further, we get:
(y^2)/256 - (x^2)/1344 = 1.
Therefore, the equation that models the hyperbolic lens is:
(y^2)/256 - (x^2)/1344 = 1.