Digital Cameras Evaluated
Does anybody like word problems???elementary/college level algebra?
1. (Remember: 1 GB= 1024 MB, 1 MB= 1024 KB, 1 KB= 1024 Bytes, 1 Byte= 8 bits)
George Lucas pioneered the use of the digital movie cameras with the most recent Star Wars film. Assume that Lucas’s camera has a swappable 80 GB hard drive that it records each pixel in x-bit color, that there are 26 frames recorded per second, and that each frame is recorded in 1600×1200 resolution.
* Create a polynomial function that gives the number of minutes of video that can be recorded before swapping in a new hard drive, as a function of x (the number of bits used to encode the color of each pixel).
*Evaluate this function for x=32 (32-bit true color).
2. Three consecutive even integers are such that the square of the third is 76 more than the square of the second. Find the three integers.
3. A rectangular parking lot is 50ft. longer than it is wide. Determine the deminsions of the parking lot if it measures 250ft diagonally.
Thank you so much for any help!
1. By the time that Star Wars was filmed, the movie film industry had already standardized on 24 frames per second and the movie was filmed in widescreen format, which by then had become standard for films for theatrical release.
Reality aside, the equation you are looking for is
t = 2^33 * c / (w * h * x * 60 * f)
where
t = capacity of each hard drive, in minutes,
2^33 = conversion constant from GB to bits,
c = the capacity of the hard drive, in GB,
w = width of each frame, in pixels,
h = height of each frame, in pixels,
60 = conversion constant from frames/sec to frames/min, and
f = frame rate, in frames/sec.
I’ll leave it to you to plug in the numbers as required.
2. Let x = the value of the second integer in the set of three even integers.
Then x – 2 = the value of the next smaller even integer, and x + 2 = the value of the next larger one.
In math notation, this becomes:
…^2 … (the square of…)
(x+2)^2… (…the third [integer]…)
(x+2)^2 = … (…is…)
(x+2)^2 = … + 76 (…76 more than…)
(x+2)^2 = …^2 + 76 (…the square of…)
(x+2)^2 = x^2 + 76 (…the second [integer].)
Now it is a question of solving for x, then plugging in to get x-2 and x+2, the other integers.
Expanding the left hand side, we get
x^2 + 4x + 4 = x^2 + 76 .
Although this looks like a quadratic equation, it turns out that when we subtract the quadratic term from one side it cancels on the other side as well, turning this into the linear equation
4x + 4 = 76
which is easily solved to get
x = 18
and
x – 2 = 16; x + 2 = 20
for the other two consecutive integers.
I could have started, as most people would, with x being the smallest integer and x+2 and x+4 being the next two in line. That would have given me the equation
(x+4)^2 = (x+2)^2 + 76
a harder equation to solve, but only slightly.
3. This is a Pythagorean Theorem problem, with the parking lot length and width forming two sides of a right triangle and the 250-foot diagonal being the hypotenuse. If we let a and b be the lot width and length, we have
a^2 + b^2 = (250 ft)^2
We also have another equation from the first sentence that we can use to get rid of one of the variables, namely:
… = … (…is…)
b = … + 50 (…longer than…)
b = a + 50 (…it is wide.)
Substituting in a + 50 everywhere b appears in the first equation then gives
a^2 + (a+50)^2 = 250^2
These problems are within the scope of elementary algebra, intermediate algebra at worst, and can be solved by a high school graduate reasonably proficient in math.
Fujifilm X100 camera evaluated for professional photographers