Solve the given problems.

The vertical cross-section of a culvert under a road is elliptical. The culvert is $18$ m wide and $12$ m high. Find an equation to represent the perimeter of the culvert with the origin at road level and $2.0$ m above the top of the culvert.

Consider steady flow of water in a downward- sloped channel of rectangular cross section. If the Row is subcritical and the flow depth is greater than the normal depth $(y > y_n)$, the flow depth will (a) increase, (b) remain constant, or (c) decrease in the flow direction.

Solve the given problems and Show some applications of straight lines.

Find the equation of the line with positive intercepts that passes through $(3, 2)$ and forms with the axes a triangle of area $12$.

Solve the exercise given below then write the correct answer.

A laser source is 2.00cm from a spherical surface of radius 3.00cm, and the laser beam is tangent to the surface. By placing the center of the sphere at the origin, and the source on the positive $x$-axis, find the equation of the line along which the beam shown in the figure is directed.

Determine the number of real solutions of the given systems of equations by sketching the indicated curves.

$$y=4 \ln x$$

$$xy=-6$$

Find the rectangular equation of each of the given polar equations. Identify the curve that is represented by the equation.

$$r \sin(\theta + \pi/6) = 3$$

Solve the given problems: sketch or display the indicated curves.

Display the graph of $r=5-4\cos\theta$ on a calculator, using $(a)$ the polar curve mode, and $(b)$ the parametric curve mode.

Water is released from a 12-m-deep reservoir into a 6-m-wide open channel through a sluice gate with a 1-m-high opening at the channel bottom. If the flow depth downstream from the gate is measured to be 3 m, determine the rate of discharge through the gate.

Solve the given problems.

The eccentricity $e$ of an ellipse is defined as $e=c/a$. A cam in the shape of an ellipse can be described by the equation $x^2+9y^2=81$. Find the eccentricity of this elliptical cam.

Find the equation of the indicated curve, subject to the given conditions. Sketch each curve.

Parabola: focus $(-3,0)$ vertex $(0,0)$

Solve the given problems.

The entrance to a building is a parabolic arch $5.6$ m high at the center and $7.4 m$ wide at the base. What equation represents the arch if the vertex is at the top of the arch?

Solve the given problems.

Determine whether the circle $x^2-6x+y^2-7=0$ crosses the $x$-axis.

Solve the given problems.

If $(a,3)$ is a point on the parabola $y=x^2+2x$, what is $a$?

Determine the type of curve from the given information.

One circular solar cell has a radius that is $2.0$ in. less than the radius $r$ of a second circular solar cell. Determine the type of curve represented by the equation relating the total area $A$ of both cells and $r$.