A professor is interested in the average length of books in her library. She has divided her books into a few different categori
2 answers:
Answer:
<h2>21 books approximately.</h2>Step-by-step explanation:
First of all we need to find the proportion of the sample. To do that, we sum all the books and then divide by the sample we need, which is 50
, because we need 50 books among 691 total.
Now, with this ratio, which is the same for all sample we would make here, we find the stratified sample of books, specifically, for sports books which are 290.
Thefore, she should take 21 sports books.
Remember that stratified sampling refers to a type of sampling method where the population is divided into separate groups, which in this case represented the type of books. Then, we use a "probability" sample, or a proportion of the sample to apply it.
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Answer: 1300 cm^3, 850 cm^2
Step-by-step explanation:
With a triangular prism, you need to calculate the surface area of the triangle first and then the length. Here's how to solve for the volume (A):
Triangles is b*h/2, plug in the equation
10*13/2
130/2
65
Now use the length of the rest of the shape, which is 20 cm.
65*20=1300
the volume is 1300 cm^3.
for B, finding the surface area requires you to analyze each individual part of the shape.
Each part colored in represents a different shape:
Red:
10*20=200
Blue:
13*10/2=65
65*2=130 (Two triangles)
Green:
13*20=260
260*2=520 (Two rectangles)
Add them all up together.
200+130+520= 850 cm^2
Answer: Choice B) Infinitely many solutions
- one solution: x = 8, y = -7/2, z = 0
- another solution: x = -12, y = 13/2, z = 10
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Explanation:
Here's the starting original augmented matrix.
We'll multiply everything in row 3 (abbreviated R3) by the value -1/4 or -0.25, which will make that -4 in the first column turn into a 1.
We use this notation to indicate what's going on:
That notation says "multiply everything in R3 by -1/4, then replace the old R3 with the new corresponding values".
So we have this next step:
Notice that the new R3 is perfectly identical to R1.
So we can subtract rows R1 and R3, and replace R3 with the result of nothing but 0's
Whenever you get an entire row of 0's, it <u>always</u> means there are infinitely many solutions.
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Now let's handle the second row. That 5 needs to turn into a 0. We can multiply R1 by 5, and subtract that from R2.
So we need to compute 5*R1-R2 and have that replace R2.
Notice that in the third column of R2, we have 9-5*2 = 9-10 = -1. So we have -1 replace the 9. In the fourth column of R2, we have 73/2 - 5*8 = -7/2. So the -7/2 replaces the 73/2.
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At this point, the augmented matrix is in RREF form. RREF stands for Reduced Row Echelon Form. It seems a bit odd that the "F" of "RREF" stands for "form" even though we say "form" right after "RREF", but I digress.
Because the matrix is in RREF form, this means R1 and R2 lead to these equations:
which simplify to
Let's get the z terms to each side like so:
Therefore, all of the solutions are of the form where z is any real number.
If z is allowed to be any real number, then we can simply pick any number we want to replace it. We consider z to be the "free variable", in that it's free to be whatever it wants. The values of x and y will depend on what we pick for z.
So the concept of "infinitely many solutions" doesn't exactly mean we can pick just <em>any</em> triple for x,y,z (admittedly it would be nice to randomly pick any 3 numbers off the top of my head and be done right away). Instead, we can pick anything we want for z, and whatever we picked, will directly determine x and y. The x and y are locked into place so to speak.
Let's say we picked z = 0.
That would lead to...
So z = 0 would lead to x = 8 and y = -7/2
Rearranging the items in alphabetical order gets us:
x = 8, y = -7/2, z = 0
We have one solution of (x,y,z) = (8, -7/2, 0)
Now let's say we picked z = 10
So we have x = -12, y = -13/2, z = 10
Another solution is (x,y,z) = (-12, 13/2, 10)
There's nothing special about z = 0 or z = 10. You can pick any two real numbers you want for z. Just be sure to recalculate the x and y values of course.
To verify each solution, you'll need to plug them back into the original equations formed by the original augmented matrix. After simplifying, you should get the same thing on both sides.