Tag: Differential Equations

Find the time until a tank with a hole is empty

by RoRi

February 23, 2016

Consider a water tank shaped like a right circular cone with its point up. Find the amount of time necessary for the tank to completely drain if there is a hole in its base. Express the time as a function of the dimension of the water tank and the area $A_0$ of the hole in the tank.

Incomplete.

Find the depth of water in a leaky tank as time goes to infinity

by RoRi

February 23, 2016

Consider a water tank with vertical sides and with cross-section a square of area 4 feet. Water exits the tank through a hole which as area equal to \frac{5}{3} square inches, and water is added to the tank at a rate of 100 cubic inches per second. Show that the water level approaches the value of $\left( \frac{25}{24} \right)^2$ feet above the hole no matter what the initial water level was.

Incomplete.

Find the time required for the water level in a tank to drop 1 foot

by RoRi

February 23, 2016

Consider a water tank with vertical sides and with cross-section a square of area 4 feet. Water exits the tank through a hole which as area equal to $\frac{5}{3}$ square inches. The water level is initially 2 feet above the level of the hole. Find the time required for the water level to drop 1 foot.

Incomplete.

Find a function whose graph has given properties

by RoRi

February 23, 2016

Let $f$ be a function such that the points $P_0 = (0,1)$ and $P_1 = (1,0)$ are on the graph of $f$ . Assume that for every point $P = (x,y)$ on the graph of $f$ , the graph is above the line segment joining $P_0$ to $P$ . Further, assume that the area $A(x)$ of the region bounded by the graph of $f$ and the line segment is $x^3$ . Find a formula for the function $f$ .

Incomplete.

Find a function whose graph has given properties

by RoRi

February 23, 2016

Let $f(x)$ be a function on the interval $[0,1]$ that is nonnegative and differentiable, and such that $f(1) = 0$ . If for each $a$ in the open interval $(0,1)$ the line given by the equation $x = a$ divides the ordinate set of $f(x)$ into regions $A$ and $B$ where $A$ denotes the leftmost region. If the areas of the two regions obey the equation

$A - B = 2f(a) + 3a + b,$

where $b$ is a constant that does not depend on $a$ , find $f(x)$ and $b$ .

Incomplete.

Find a function whose ordinate set generates a solid of revolution with volume x² f(x)

by RoRi

February 23, 2016

Let $f(x)$ be a nonnegative, differentiable function whose graph passes through both points $(0,0)$ and $\left( 1, \frac{2}{\pi}\right)$ . For every real number $x > 0$ , the ordinate set of $f$ on the interval $[0,x]$ generates a solid of revolution when rotated about the $x$ -axis whose volume is given by

$x^2 f(x).$

Find the formula for the function $f(x)$ .

Incomplete.

Find a function which divides a rectangle into pieces with given properties

by RoRi

February 23, 2016

Consider a curve whose Cartesian equation is given by $y = f(x)$ , and which passes through the origin. A rectangular region is drawn with one corner at the origin, and the other corner on the curve of the graph of $f(x)$ . The curve $f(x)$ then divides the rectangle into two pieces $A$ and $B$ . These two pieces of the rectangle then generate solids of revolution when rotated about the $x$ -axis. If the volume of one solid of revolution is always $n$ times the volume of the other solid of revolution, find the equation for $f(x)$ .

Incomplete.

Find a function which divides a rectangle into pieces with given properties

by RoRi

February 23, 2016

Consider a curve whose Cartesian equation is given by $y = f(x)$ , and which passes through the origin. A rectangular region is drawn with one corner at the origin, and the other corner on the curve of the graph of $f(x)$ . The curve $f(x)$ then divides the rectangle into two pieces $A$ and $B$ such that one of the regions has area $n$ times the area of the other for every such rectangle. Find the equation of $f(x)$ .

Incomplete.

Find a Cartesian equation for the pursuit path for given parameters

by RoRi

February 23, 2016

Let Q be a point which moves upward along the positive y-axis and let P be a point which starts at (1,0) and pursues Q according to an equation which stipulates that the distance from P to the y-axis is exactly \frac{1}{k} times the distance from Q to the origin for $k$ a positive number. Find the Cartesian equation for the path of pursuit the point P traces out.

Incomplete.

Find a Cartesian equation for the pursuit path of given parameters

by RoRi

February 23, 2016

Let $Q$ be a point which moves upward along the positive $y$ -axis and let $P$ be a point which starts at $(1,0)$ and pursues $Q$ according to an equation which stipulates that the distance from $P$ to the $y$ -axis is exactly $\frac{1}{2}$ the distance from $Q$ to the origin. Find the Cartesian equation for the path of pursuit the point $P$ traces out.

Incomplete.