The function f(x) = 28(1.45)x models the number of cats and dogs adopted from a local animal shelter, where x represents the num
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<h3>Answer:</h3>
- 8 pieces
- 3/8 foot
a) Let n represent the number of 1 7/8 ft pieces. Then we have ...
... 16 1/2 ft = n × (1 7/8 ft)
We can find n by dividing this equation by 1 7/8.
... (16 1/2)/(1 7/8) = n = (132/8)/(15/8) = 132/15
... n = 8 12/15 = 8 4/5
The integer number of pieces that can be cut is 8 pieces.
b) The length of the left-over piece is 4/5 of the full length of a piece, so is ...
... (4/5) × (15/8 ft) = 12/8 ft
We want to divide this into 4 equal lengths, so each of those lengths will be 1/4 of the length of the left-over:
... (1/4) × (12/8 ft) = 3/8 ft
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<em>Comment on arithmetic with fractions</em>
This problem is easier if you <em>convert the mixed numbers to improper fractions</em>. This is done by expressing the integer as a number of fractional parts, then adding the fraction part of the mixed number:
... a + b/c = (a×c/c) + b/c = (ac +b)/c
Division with fractions is taught a couple of different ways. One way you can do it is to "invert and multiply". That is, you multiply the numerator fraction by the reciprocal of the denominator fraction:
... (a/b)/(c/d) = (a/b)×(d/c)
Another way you can do the division is to express each of the numerator and denominator fractions over the same denominator. Then the quotient is the ratio of numerators.
... (a/c)/(b/c) = a/b
This is the method we used here. Effectively, we converted 16 1/2 to 16 4/8 then to (16×8 +4)/8 = 132/8. Then the numerator and denominator fractions both had denominators of 8, so (132/8)/(15/8) = 132/15. This fraction can be reduced by removing a factor of 3 from numerator and denominator to give ...
... 132/15 = 44/5 = 8 4/5
Short answer 2 * sec^2(theta)
Find the common denominator
The common denominator is (1 - sin(theta) * (1 + sin(theta) )
Multiply each fraction's numerator and denominator by the common denominator
Remove the brackets and cancel the sin(theta) terms in the numerator.
Combine the denominator
2 / (1 - sin^2(theta)) but 1 - sin^2(theta) = cos^2(theta)
2 / (cos^2 (theta) but 1/cos^2(theta) = sec^2(theta)
2 * sec^2 (theta) <<<<< Answer
Find the common denominator
The common denominator is (1 - sin(theta) * (1 + sin(theta) )
Multiply each fraction's numerator and denominator by the common denominator
Remove the brackets and cancel the sin(theta) terms in the numerator.
Combine the denominator
2 / (1 - sin^2(theta)) but 1 - sin^2(theta) = cos^2(theta)
2 / (cos^2 (theta) but 1/cos^2(theta) = sec^2(theta)
2 * sec^2 (theta) <<<<< Answer