Answer:
The dimensions of the smallest piece that can be used are: 10 by 20 and the area is 200 square inches
Step-by-step explanation:
We have that:

Let the dimension of the paper be x and y;
Such that:


So:

Substitute 128 for Area

Make x the subject

When 1 inch margin is at top and bottom
The length becomes:


When 2 inch margin is at both sides
The width becomes:


The New Area (A) is then calculated as:

Substitute
for x

Open Brackets

Collect Like Terms



To calculate the smallest possible value of y, we have to apply calculus.
Different A with respect to y

Set

This gives:

Collect Like Terms

Multiply through by 


Divide through by 2

Take square roots of both sides



Recall that:



Recall that the new dimensions are:


So:




To double-check;
Differentiate A'




The above value is:

This means that the calculated values are at minimum.
<em>Hence, the dimensions of the smallest piece that can be used are: 10 by 20 and the area is 200 square inches</em>
<h3>
Answer: A. 18*sqrt(3)</h3>
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Explanation:
We'll need the tangent rule
tan(angle) = opposite/adjacent
tan(R) = TH/HR
tan(30) = TH/54
sqrt(3)/3 = TH/54 ... use the unit circle
54*sqrt(3)/3 = TH .... multiply both sides by 54
(54/3)*sqrt(3) = TH
18*sqrt(3) = TH
TH = 18*sqrt(3) which points to <u>choice A</u> as the final answer
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An alternative method:
Triangle THR is a 30-60-90 triangle.
Let x be the measure of side TH. This side is opposite the smallest angle R = 30, so we consider this the short leg.
The hypotenuse is twice as long as x, so TR = 2x. This only applies to 30-60-90 triangles.
Now use the pythagorean theorem
a^2 + b^2 = c^2
(TH)^2 + (HR)^2 = (TR)^2
(x)^2 + (54)^2 = (2x)^2
x^2 + 2916 = 4x^2
2916 = 4x^2 - x^2
3x^2 = 2916
x^2 = 2916/3
x^2 = 972
x = sqrt(972)
x = sqrt(324*3)
x = sqrt(324)*sqrt(3)
x = 18*sqrt(3) which is the length of TH.
A slightly similar idea is to use the fact that if y is the long leg and x is the short leg, then y = x*sqrt(3). Plug in y = 54 and isolate x and you should get x = 18*sqrt(3). Again, this trick only works for 30-60-90 triangles.
Answer:
I'd say d but if you worry choose to be then go with B
Answer:
Hi, There! Leah Is Correct
Step-by-step explanation:
A fraction is a repeating decimal if the prime factors of the denominator of the fraction in its lowest form do not only contain 2s and/or 5s or do not have any prime factors at all.
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-Darkspirit-
I believe the answer is 4 because 6m/29= 20.6897 and so then you do 20.6897 x 3= 62.0691. So 3 is too small so do 20.6897x 4= <span>82.7588. 82 >78 so the answer is that you need 4 boards.
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