Final answer:
Tunnel A is represented by a circle with the equation (x + 14)² + y² = 144 in standard form, having a center at (-14, 0) and a radius of 12 feet, which is spacious enough for a truck 8 feet wide and 13.5 feet high to pass through.
Step-by-step explanation:
To write the equation for Tunnel A in standard form, we need to complete the square for both the x and y terms in the equation x² + y² + 28x + 52 = 0. Completing the square for the x-terms involves taking half of the coefficient of x, which is 14, squaring it, and adding it to both sides of the equation. Since this is a circle, we do not have a y-term to complete, but if we had one, we would follow a similar process for the y-term.
First, we rewrite the equation and then complete the square:
x² + 28x + y² = -52
x² + 28x + 14² + y² = 14² - 52
(x + 14)² + y² = 196 - 52
(x + 14)² + y² = 144
Now that we have completed the square, the equation is in standard form:
(x + 14)² + y² = 144
Comparing this with the standard form of a circle equation (x - h)² + (y - k)² = r², we can see that the conic section of Tunnel A is a circle with a center (h, k) = (-14, 0) and radius r = 12 feet. To determine if the truck can pass through the tunnel, we need to consider the diameter of the tunnel is twice the radius, which is 24 feet, so a truck that is 8 feet wide and 13.5 feet high can comfortably pass through Tunnel A.