The area of a rectangular photograph is 59 square inches.It is to be mounted on a rectangular card with a boarder of 1 inch at each side and 2 inches at the top and bottom.Express the total area of the photograph and the boarder,A, as a function of the width of the photograph,x.what width produces the maximum area
let p = area of photo = 59
let x be the width. So, the height is 59/x
m = (x+2)(59/x + 4)
m = 59 + 118/x + 4x + 8
to find max mounted area, set m' = 0
m' = -118/x² + 4 = (4x² - 118)/x^2
So, m' = 0 when 4x²-118 = 0
x² - 29.5 = 0
x = 5.43"
Use 'Getting it out of his system.' Cartoon comment on the declining influence of Senator Joseph McCarthy and his anti-Communist crusade within the Republican Party. Cartoon by Edmund Valtman, 11 March 1954 to answer the question.
What does the image BEST suggest about McCarthyism?
A. It was a destructive trend that had run through the Republican party.
B. There was no place for McCarthyism in postwar world.
C. McCarthyism was scorned by both Democrats and Republicans.
D. Democrats were immune to the problems associated with Mcarthy's actions
A. It was a destructive trend that had run through the Republican party.
To find the total area of the photograph and the border as a function of the width of the photograph, x, we need to consider the dimensions of the border.
Given:
Area of the rectangular photograph = 59 square inches.
Border width:
- 1 inch on each side
- 2 inches at the top and bottom
Let's calculate the total width and height of the photograph, considering the borders.
Width including the borders = x + 2(1) = x + 2
Height including the borders = x + 2(2) = x + 4
The total area, A, of the photograph and borders can be expressed as:
A = (width including borders) * (height including borders)
Substituting the values we calculated:
A = (x + 2) * (x + 4)
To find the width that produces the maximum area, we need to determine the value of x that maximizes the function A = (x + 2) * (x + 4).
We can achieve this by finding the vertex of the quadratic function or by applying calculus.
Method 1: Calculus
To find the maximum area, we differentiate the function A with respect to x, set it equal to zero, and solve for x.
Method 2: Completing the Square
We can rewrite the function A in the form of a perfect square trinomial by completing the square.
Let's use Method 2 to find the width, x, that produces the maximum area.
A = (x + 2) * (x + 4)
= x^2 + 2x + 4x + 8
= x^2 + 6x + 8
Now, let's complete the square:
A = (x + 3)^2 - 1
Since the area, A, cannot be negative, the maximum area occurs when (x + 3)^2 = 1.
Taking the square root of both sides, we have:
(x + 3) = ±1
Solving for x:
x + 3 = 1 or x + 3 = -1
If x + 3 = 1, then x = 1 - 3 = -2.
If x + 3 = -1, then x = -1 - 3 = -4.
Since we are considering the width of a photograph, x must be positive. Therefore, the width that produces the maximum area is x = -2.
However, note that in this context, a width of -2 inches is not feasible. The maximum area is achieved when the width of the photograph is 2 inches.