Running head: RADICAL FORMULAS
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Real World Radical Formulas
John Q. Student
MAT 222 Week 3 Assignment
Real World Radical Formulas (title required on first line)
Though radicals seem complicated at first glance, the concept simply extends what is
already known about exponents and the order of operations. Additionally, manipulating formulas
that include radicals is no different than those without, providing appropriate rules are followed.
These rules include accurately finding the cube and square root for numbers and understanding
the application of the solution in the real world.
It is stated in #103 on page 605 (Dugopolski, 2012) that the capsize screening value C
should be less than 2 if a boat is to be considered safe for ocean sailing. The formula is given as
C = 4d-1/3b where d is the displacement in pounds and b is the beam width in feet. The exponent
of -1/3 means that the cube root of d will be taken and then the reciprocal of that number will be
used in the multiplication.
a) For the first example, a 1963 Cascade 29 sailboat’s capsize screening value will be
calculated. The boat has a beam of 8 ft 2 in. and a displacement of 8500 lbs. The inches value
will be converted into a decimal by division (2/12).
C = 4d-1/3b
C = 4(8500)-1/3(8.17) Here are the values plugged into the formula. According to
order of operations, the exponents are solved first (exponent computed by calculator).
C = 4(.049)(8.17) Now it is just two multiplications.
C = .196(8.17)
C = 1.60
The capsize screening value is less than 2.
b) Here is a formula similar to the capsize screening value. D = 1.1g -1/4H. The formula
will be solved for g.
D = 1.1g -1/4H No substitution is needed because the formula is being manipulated.
D = 1.1g-1/4H
Divide both sides by 1.1H.
Raise all parts of both sides to the -4th power . The (1.1H)-4
= ( g-1/4)-4
on the left side becomes its reciprocal to the 4th power.
The right side becomes simply g.
1.4641H4 = g
The formula has now been solved for g.
Problem 104 on page 606 (Dugopolski, 2012) gives a formula for computing the sail
power of a boat based on the area of the sail A in square feet, and the displacement of d pounds.
The formula is S = 16Ad-2/3 where S stands for the sail area-displacement ratio. The -2/3
exponent on the d means first the cube root will be taken, then the result of that will be squared,
and finally the last result will divide 16A (i.e. sent to the denominator).
a) The boat has a sail area of 510 square feet and a displacement of 8500 lbs.
S = 16Ad-2/3
S = 16(510)(8500)-2/3
Values plugged into the formula. Exponents are calculated
first (exponent computed by calculator).
S = 16(510)(.0024)
Complete multiplication left to right.
S = 8160(.0024)
S = 19.584
This seems to be a good amount of sail power for this boat. It
might be so high because the replaced original sail was only 405 square feet with a larger sail of
b) Here is a formula similar to the sail area-displacement ratio: T = 8Bg -3/4 . Once again
the formula will be solved for g.
T = 8Bg -3/4
Divide both sides by 8B.
Now let us consider an analysis of the graph for the capsize screening value, C.
C = 4d-1/3b.
Since there are three variables this function actually represents a family of functions based on
varying values for the boats beam, b. Say we want to look at the graphs for beams of 8, 10 and
12 feet. That means 3 functions, one for each value of b.
C = 4d-1/3*8 = 32d-1/3
C = 4d-1/3*10 = 40d-1/3
C = 4d-1/3*12 = 48d-1/3
Here is the graph of the family of functions for b = 8 (purple), 10 (black) and 12 (red) foot
Recall that the boat is less likely to capsize when the capsize screening factor, C, is less than
2. These graphs can now be used to determine the safe displacements given a particular beam.
Note that a larger beam can carry a larger displacement and still remain safe.
Conclusion paragraph would go here. Remember to include 2-3 sentences to make a
Dugopolski, M. (2012). Elementary and intermediate algebra (4th ed.). New York, NY:
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