# .There was a proposal for the new size of the advisory boards to be placed at strategic locations along the expressway. The board must contain 50 square meters of print and margins of 4 meters at the top and bottom while 3 meters on each side.?

##
a. Find a mathematical model expressing the total area of the board as a function of the horizontal dimension of the region covered by the printed material.

b. Determine the domain of the function.

c. Prove that the function is continuous on its domain.

d. Sketch the graph of the function obtained in (a).

e. Determine (to the nearest meter) the dimensions of the smallest board that meets the specifications.

a. Find a mathematical model expressing the total area of the board as a function of the horizontal dimension of the region covered by the printed material.

b. Determine the domain of the function.

c. Prove that the function is continuous on its domain.

d. Sketch the graph of the function obtained in (a).

e. Determine (to the nearest meter) the dimensions of the smallest board that meets the specifications.

To the nearest

Let us set up the following variables:

# {(x, "Printed width of poster (m)"), (y, "Printed height of poster (m)"), (A, "Total Area of the board (m"^2")") :} #

Part (A):

Then the dimensions of the entire board are:

# {("Width", =x+3+3,=x+6), ("Height", =y+4+4,=y+8) :} #

And so the total Area of the board is given by:

# A=(x+6)(y+8) #

But we also know that the print area is

# xy = 50 => y=50/x#

Thus by substitution into the earlier result, we have:

# A = (x+6)(50/x+8) #

# \ \ \ = x(50/x+8) + 6(50/x+8)#

# \ \ \ = 50+8x+300/x + 48 #

# \ \ \ = 8x+300/x + 98 #

Part (B):

The function is defined for all values

# { x in RR | x != 0 } #

Given the context of the problem, we cannot have a negative length, so in reality we have a domain:

# { x in RR | x gt 0 } => x in (0,oo)#

Part (C):

An Intuitive "proof": The function is defined for all values

Part (D):

graph{y=8x+300/x + 98 [-5, 30, -50, 500]}

Part (E):

For maximum or minimum, we seek a critical point of the area function,

# f'(x) = 8 -300/x^2 #

At a critical point, the first derivative vanishes, thus we require:

# f'(x) = 0 => 8 -300/x^2 = 0 #

# :. 300/x^2 = 8 #

# :. x^2 = 300/8 #

# :. x = +-sqrt(300/8) = +- (5sqrt(6))/2 #

We discard the negative solution (due to the domain restriction), thus we gain a single solution

# x = (5sqrt(6))/2 ~~ 6.1 \ m#

Which gives us:

# A(x) = 8*(5sqrt(6))/2 + 300/((5sqrt(6))/2) + 98 #

# \ \ \ \ \ \ \ \ = 20sqrt(6)+20sqrt(6)+98 #

# \ \ \ \ \ \ \ \ = 98 + 40sqrt(6) \ m^2#

# \ \ \ \ \ \ \ \ ~~ 196.0 \ m^2#

We can visually confirm from the graph, that the solution is both consistent and corresponds to a minimum, and the dimensions of the board sought are thus:

# {("Width", =(5sqrt(6))/2+6, ~~ 12.12 \ m), ("Height", =(50*2)/(5sqrt(6))+8, ~~ 16.16 \ m) :} #

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- How do you find the linearization of #F(x) = cos(x)# at a=pi/4?
- The corners are removed from a sheet of paper that is #3# ft square. The sides are folded up to form an open square box. What is the maximum volume of the box?
- What is the local linearization of #e^sin(x)# near x=1?
- An arched window (an upper semi-circle and lower rectangle) has a total perimeter of #10 \ m#. What is the maximum area of the window?
- What is the rate of change of the width (in ft/sec) when the height is 10 feet, if the height is decreasing at that moment at the rate of 1 ft/sec.A rectangle has both a changing height and a changing width, but the height and width change so that the area of the rectangle is always 60 square feet?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7