Answer:
A. Circle.
[tex](x+15)^2+(y-0)^2=13^2[/tex]
B. Parabola.
[tex](x-15)^2=-16(y-20)[/tex]
C. Maximum height of Tunnel A = 13 ft.
Maximum height of Tunnel B = 20 ft.
The truck can only pass through Tunnel B without damage.
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{6.3cm}\underline{General equation for any conic section}\\\\$Ax^2+Bxy+Cy^2+Dx+Ey+F = 0$\\\\where $A, B, C, D, E, F$ are constants.\\\end{minipage}}[/tex]
Circle: A and C are non-zero and equal, and have the same sign.
Ellipse: A and C are non-zero and unequal, and have the same sign.
Parabola: A or C is zero.
Hyperbola: A and C are non-zero and have different signs.
Tunnel A
[tex]x^2+y^2+30x+56=0[/tex]
As the coefficients of x² and y² are non-zero, equal and have the same sign, the conic section is a circle.
[tex]\boxed{\begin{minipage}{4 cm}\underline{Equation of a circle}\\\\$(x-h)^2+(y-k)^2=r^2$\\\\where:\\ \phantom{ww}$\bullet$ $(h, k)$ is the center. \\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]
Rewrite the given equation for Tunnel A in the standard form of the equation of a circle:
[tex]\implies x^2+y^2+30x+56=0[/tex]
[tex]\implies x^2+30x+y^2-56[/tex]
[tex]\implies x^2+30x+\left(\dfrac{30}{2}\right)^2+y^2=-56+\left(\dfrac{30}{2}\right)^2[/tex]
[tex]\implies x^2+30x+225+y^2=-56+225[/tex]
[tex]\implies (x+15)^2+(y-0)^2=169[/tex]
[tex]\implies (x+15)^2+(y-0)^2=13^2[/tex]
Therefore, the center of the circle is (-15, 0) and the radius is 13.
Tunnel B
[tex]x^2-30x+16y-95=0[/tex]
There is no term in y² so the coefficient of y² is zero. Therefore, the conic section is a parabola.
[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Standard form of a parabola}\\(with a vertical axis of symmetry)\\\\$(x-h)^2=4p(y-k)$\\\\where:\\ \phantom{ww}$\bullet$ $p\neq 0$. \\ \phantom{ww}$\bullet$ $(h,k)$ is the vertex.\\\end{minipage}}[/tex]
Rewrite the given equation for Tunnel B in the standard form a parabola:
[tex]\implies x^2-30x+16y-95=0[/tex]
[tex]\implies x^2-30x=-16y+95[/tex]
[tex]\implies x^2-30x+\left(\dfrac{30}{2}\right)^2=-16y+95+\left(\dfrac{30}{2}\right)^2[/tex]
[tex]\implies x^2-30x+225=-16y+95+225[/tex]
[tex]\implies (x-15)^2=-16(y-20)[/tex]
Therefore, the vertex is (15, 20).
Maximum height of Tunnel A
The maximum point of a circle is the sum of the y-value of its center and its radius:
Maximum height of Tunnel B
The maximum point of a downwards opening parabola is the y-value of its vertex:
As the truck is 13.5 feet high, it cannot pass through Tunnel A since the maximum height of Tunnel A is 13 feet.
The maximum height of Tunnel B is certainly adequate for the truck to pass through. However, to determine if the truck can pass through Tunnel B safely, we also need to find the width of the tunnel when its height is 13.5 feet. To do this, find the x-values of the parabola when y = 13.5. If the difference in x-values is 8 or more, then the truck can pass through safely.
Substitute y = 13.5 into the equation for Tunnel B and solve for x:
[tex]\implies (x-15)^2=-16(13.5-20)[/tex]
[tex]\implies (x-15)^2=-16(-6.5)[/tex]
[tex]\implies (x-15)^2=104[/tex]
[tex]\implies \sqrt{(x-15)^2}=\sqrt{104}[/tex]
[tex]\implies x-15=\pm\sqrt{104}[/tex]
[tex]\implies x=15\pm\sqrt{104}[/tex]
Now find the difference between the two found values of x:
[tex]\implies (15+\sqrt{104})-(15-\sqrt{104})[/tex]
[tex]\implies 15+\sqrt{104}-15+\sqrt{104}[/tex]
[tex]\implies 2\sqrt{104}[/tex]
[tex]\implies 20.39607...[/tex]
Therefore, as the width of Tunnel B is 20.4 ft when its height is 13.5 ft, the 8 ft wide truck can easily pass through without damage since 20.4 ft is greater than the width of the truck.