A tunnel has a shape of a semi ellipse that is feet at the center and 36 feet wide across at the base. At most how high should a passing truck be, if it is 12 feet wide, for it to be able to fit through the tunnel? Round off your answer to two decimal places?
You neglected to provide the center height. But, just for the sake of the exercise, let's say it is 15 feet high. That means
a = 18
b = 15
and the equation of the ellipse is
x^2/324 + y^2/225 = 1
So now you want to find y when x = 6
y < 10√2 = 14.1 ft
To determine the maximum height a passing truck can be to fit through the tunnel, we need to use the concept of similar triangles.
The semi-ellipse shape of the tunnel can be represented as an isosceles triangle and a semicircle joined together. The width of the tunnel at the base is given as 36 feet. This means the width of each half of the ellipse (isosceles triangle) is 36/2 = 18 feet.
Let's assume the height of the passing truck is "h" feet.
Since the truck is 12 feet wide, it should fit within the width of the tunnel, which is 18 feet across the base. Therefore, we can set up a proportion:
12 ft / 18 ft = h ft / r ft
Where "r" represents the radius of the semicircle, and "h" represents the height of the truck.
For the proportion, the radius of the semicircle is half the width of the tunnel, so r = 18/2 = 9 feet. Substituting these values into the proportion, we have:
12 ft / 18 ft = h ft / 9 ft
To find h, we can cross-multiply and solve for h:
12 ft * 9 ft = 18 ft * h ft
108 ft^2 = 18 ft * h ft
Divide both sides by 18 ft:
6 ft = h ft
Therefore, the maximum height of the passing truck should be 6 feet to fit through the tunnel.
To find the maximum height that a passing truck can have to fit through the tunnel, we need to determine the height at the center of the semi ellipse.
First, we can find the semi-major axis (a) of the semi ellipse by dividing the width at the base (36 feet) by 2. Therefore, a = 36/2 = 18 feet.
Next, we can find the semi-minor axis (b) of the semi ellipse. This can be calculated using the formula:
b = sqrt(a^2 - c^2), where c is the distance from the center to one end of the semi-major axis.
In this case, c = 12 feet, as the truck is 12 feet wide.
Plugging in the values:
b = sqrt(18^2 - 12^2)
b = sqrt(324 - 144)
b = sqrt(180)
b ≈ 13.42 feet (rounded to two decimal places)
Therefore, the maximum height of the truck should be 13.42 feet (rounded to two decimal places) to fit through the tunnel.