a rectangle PQRS with the length v-cm and the width z-cm and its perimeter 88cm. the rectangle is axis of symmetry and parallel to RS to generate cylinder. find the maximum value of its volume
Draw the rectangle with vertices at P=(0,0) Q=(v,0) R=(v,z) S = (0,z)
The axis of symmetry is the line y = z/2
We have v+z = 44
You don't say how PQRS is rotated, but I assume it is around the axis of symmetry, so its volume is
v = π(z/2)^2 v = π(z/2)^2(44-z) = π/4 (44z^2 - z^3)
dv/dz = π/4 (88z - 3z^2)
dv/dz = 0 when z = 88/3
so max v = 85184π/27
how did you get 88/3? and 85184n? don't get it
To find the maximum volume of the cylinder generated by the rectangle, we need to set up an equation and differentiate it to find the maximum point.
Let's start by visualizing the rectangle PQRS and the cylinder it generates.
R---------S
/ \
/ \
P---------------Q
The rectangle has sides of length v cm and z cm. The perimeter of the rectangle is given as 88 cm. Since it is axis-symmetric and parallel to RS, the length PQ is equal to RS, and the width QR is equal to PS.
We can set up the equation using the perimeter:
Perimeter of the rectangle = 2 * (Length + Width) = 2 * (v + z) = 88 cm
Simplifying the equation, we have:
v + z = 44 ----(Equation 1)
Now, let's consider the cylinder generated by this rectangle. To do this, we will consider the rectangle as the base of the cylinder and the length v as the height of the cylinder.
The volume of a cylinder is given by the formula:
Volume of cylinder = π * r^2 * h
In our case, since the rectangle is axis-symmetric, the radius of the cylinder is given by half the width (QR). Thus:
r = QR/2 = z/2 ----(Equation 2)
Substituting r and v into the volume equation, we get:
Volume of cylinder = π * (z/2)^2 * v
= (π * v * z^2) / 4
Now, we need to maximize this volume. To do this, we can differentiate the volume equation with respect to z, set the derivative equal to zero, and solve for z.
Differentiating the volume equation with respect to z:
d/dz [(π * v * z^2) / 4] = (π * v * 2z) / 4
= (π * v * z) / 2
Setting the derivative equal to zero:
(π * v * z) / 2 = 0
Since v is a non-zero value, we can divide both sides of the equation by (π * v / 2):
z = 0
We have found that z = 0 satisfies the condition for a maximum volume when considering only the derivative. However, upon observing the rectangle and the cylinder, we can see that z cannot be zero because it represents the width of the rectangle. Therefore, this is not a valid solution.
Instead, we look back to Equation 1:
v + z = 44
By applying the constraint of the problem statement, which states that the rectangle is parallel to RS, we know that v = RS.
Substituting v = RS into the equation, we get:
RS + z = 44
Rearranging the equation:
RS = 44 - z ----(Equation 3)
Now we have two equations that relate z and RS:
Equation 2: r = z/2
Equation 3: RS = 44 - z
We need to use these equations to express the volume in terms of a single variable (z) to differentiate it.
From Equation 3, we can solve for RS:
RS = 44 - z
Using RS as the length in the cylindrical volume equation, we have:
Volume of cylinder = (π * (z/2)^2 * (44 - z)) / 4
= π * z^2 * (44 - z) / 16
Now we have the volume equation in terms of a single variable, z.
To find the maximum value of the volume, we can differentiate this volume equation with respect to z, set the derivative equal to zero, and solve for z.
Differentiating the volume equation with respect to z:
d/dz [π * z^2 * (44 - z) / 16] = (π / 16) * (2z * (44 - z) - z^2)
Setting the derivative equal to zero:
(π / 16) * (2z * (44 - z) - z^2) = 0
Now, we solve the equation for z.
2z * (44 - z) - z^2 = 0
(2z - z^2) * (44 - z) = 0
There are two possibilities for this equation to hold true:
1. 2z - z^2 = 0
=> z(2 - z) = 0
=> z = 0 or z = 2
2. 44 - z = 0
=> z = 44
However, z = 2 and z = 44 do not satisfy the condition of the problem statement that the rectangle's width is z cm. Therefore, the only valid solution is z = 0.
But z = 0 implies that the width of the rectangle is zero, which is not possible. Therefore, there is no maximum volume for the cylinder generated by the given rectangle.