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When a fighter jet travels faster than sound, it generates a sonic boom shock wave in the shape of a cone. If the plane is flying at a constant altitude, this cone intersects the ground in the shape of one branch of a hyperbola. Suppose a jet is flying north at level altitude and at a speed of 500 m/s (which is a supersonic speed). In this model, let the position of the jet be the origin. The vertex of the sonic boom hyperbola is on the ground 15 km behind the jet, and the hyperbola has an eccentricity of 1.5. Use a scale of 1 unit 1 km for this model. 21. Find the coordinates of the focus and the equations of the asymptotes. Round all answers to the nearest hundredth. focus: asymptotes: 22. Write an equation for the sonic boom hyperbola in standard form.

Solution

21.

eccentricity of the hyperbola is e = c/a =1.5

and c^2 = a^2 + b^2 , where a and b are the length of the semi major and minor axis respectively.

the vertex of the sonic boom hyperbola will be (-15 , 0)

the plane is flying at the same altitude so the y coordinate will remain the same and since the vertex is 15 km behind th jet {remember the jet is at the orign that is (0,0)} so the vertex will be (-15 , 0)

here the vertex is 15 km on either side of the so a = 15

now e = 1.5 = c/a = c/15

=> c = 22.5

now the coordinates of the focus are given by

f = (h + c , k) and (h - c , k)

here (h,k) is the center of the sonic boom hyperbola and that is the orign

=> h = 0 and k = 0

so the focii are (22.5 , 0) and (-22.5, 0)

the sonic boom hyperbola is on the negative side of the orign so we are dealing with the negative part of the hperbola

=> the coordinate of the focus is f = ( - 22.5 , 0) or ( -23 , 0) nearest hundereth

the equation of the symptotes could be found by the equation

y = +- b/a*(x - h) + k

we need to find b first

c^2 = a^2 + b^2

=> b = sqrt(c^2 - a^2) = sqrt(23^3 - 15^2 ) = 109.28 = 109

y = +- 109/15(x - 0) + 0

hence the equation of the asymptotes are:

y = 109x/15   and   y = - 109x/15

22.

equation of the sonic boom hyperbola in the standard form is :

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

(h,k) = (0,0) and a = 15 and b = 23

=>

x^2/15^2 - y^2/23^2 = 1

=> x^2/225 - y^/529 = 1          --------> equation of the hyperbola