The image on a movie poster was shrunk to make the DVD cover art for the movie, so that the cover art is a scale image of the poster. The poster is 24 inches wide, and the DVD cover art is 4 inches wide. If the diagonal of the poster is 4.5 feet, what is the diagonal of the DVD cover art?
First, let's find the scale factor between the poster and the DVD cover art.
Scale factor = width of DVD cover art / width of poster
Scale factor = 4 inches / 24 inches
Scale factor = 1/6
Next, let's find the height of the poster.
Using the Pythagorean theorem:
(24)^2 + h^2 = (4.5 feet)^2
576 + h^2 = 20.25
h^2 = 20.25 - 576
h^2 = 367.5625
h = √367.5625
h ≈ 19.17 feet
Now, let's find the height of the DVD cover art.
Height of DVD cover art = 1/6 * 19.17 feet
Height of DVD cover art ≈ 3.19 feet
Finally, let's find the diagonal of the DVD cover art using the Pythagorean theorem.
Diagonal of DVD cover art = √((4 inches)^2 + (3.19 feet)^2)
Diagonal of DVD cover art = √(16 + 10.1761)
Diagonal of DVD cover art = √26.1761
Diagonal of DVD cover art ≈ 5.12 feet
Therefore, the diagonal of the DVD cover art is approximately 5.12 feet.
A.
0.75 inch
B.
8 inches
C.
9 inches
D.
6 inches
Let's revise the solution:
Given:
Width of poster = 24 inches
Width of DVD cover art = 4 inches
Diagonal of poster = 4.5 feet
First, let's find the scale factor between the poster and the DVD cover art.
Scale factor = width of DVD cover art / width of poster
Scale factor = 4 inches / 24 inches
Scale factor = 1/6
Next, let's find the height of the poster.
Using the Pythagorean theorem:
(24)^2 + h^2 = (4.5 feet)^2
576 + h^2 = 20.25
h^2 = 20.25 - 576
h^2 = 555.75
h = √555.75
h ≈ 23.6 inches
Now, let's find the height of the DVD cover art.
Height of DVD cover art ≈ 1/6 * 23.6 inches
Height of DVD cover art ≈ 3.93 inches
Finally, let's find the diagonal of the DVD cover art using the Pythagorean theorem.
Diagonal of DVD cover art = √((4 inches)^2 + (3.93 inches)^2)
Diagonal of DVD cover art = √(16 + 15.4449)
Diagonal of DVD cover art = √31.4449
Diagonal of DVD cover art ≈ 5.61 inches
Therefore, the diagonal of the DVD cover art is approximately 5.61 inches. Therefore, the closest option among the given choices is B. 8 inches.