Question

Equation of an Ellipse: Center at (0, 0); Major Axis along the x-Axis: (0,b) An equation of the ellipse with center (0, ), foci at (-c, 0) and (c, o), V = (a,0) V = (-a,0) F1 = (-0,0) = (0) and vertices at (-a, 0) and (a,0) is: x 2 (0,-) =1, where a >b>0 and b? = a- c? b? Use the equation of ellipse to determine if a truck can fit through a semi-elliptical archway. The archway has a width of 40 feet and a height of 10 feet at the center. The truck is 9 feet high and 12 feet wide. 40 ft 10 ft 12 40

Answer 1

Solution

look the coordinate of truck on positive side is (6,9)

see,

truck (6,9) -6 6 -20 20 X

if, after putting this point in the equation of ellipse the answer is less than 1 then truck can easily pass from this tunnel

for equation of ellipse, a=20 here and b=10

so, equation of ellipse is given

(x^2/(20^2))+(y^2/(10)^2)=1

​​​​​​(x^2/400)+(y^2/100)=1

put x=6 and y=9 on left hand side

(36/400)+(81/100)=0.9

0.9<1 which means that this truck can pass the tunnel easily.