Real World Radical Formulas
Radical formulas are used in many ways to solve real word problems. Even with complex equation, it can still be solved if some of variables are known. Radical formulas are used to solve for safety factors of real life situation and for our case, the sailboat stability. Though radicals may seem complicated, same rules applied in exponents are used bearing in mind the effect of negative power. Product rule, power rule and quotient rule are still used in radical formulas (Dugopolski, 2012).
From the given problem, #103 on page 605 (Dugopolski, 2012), the capsizing value C should be less than 2 if the boat should be considered safe for sailing. The sailing equation is given as C = 4d-1/3b where d, the radical, 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. Part a requires the determination of capsize screening, C given the value of d = 23245 and b =13.3
Substituting the values of b and d in the above equation.
C =4 x 23245-1/3 x13.5
=4 x1/(cube root 23245) x 13.5
From the calculator, 3rd root of 23245 = 28.54 (rounded to two decimal place)
= 4 x 1/28.54 x13.5
= 4 x 0.035 x 13.5 from calculator, 1/ 28.54 = 0.035
Hence the value of C = 1.89
Part b asks to solve the value of d from the formula. C = 4d-1/3b
Since we are asked the value of d, we can make it the subject of the formula from the original equation. Making d the subject of the formula means that, we equate it to the other constants.
First divide both sides of the equation with 4b in order to eliminate them from right hand side
C/4b = ( 4d-1/3b)/4b
C/4b = d-1/3
Secondly we need to take care of the negative power and the root. In doing this, raise each of the three terms with exponent -3
C -3/(4b)-3 = d-1/3x-3 the power rule is used
C -3/(4b)-3 = d
(4b)3/C3 = d
64b3/C3 = d
Since we have made d the subject of the formula and the values of C and b, are given, we can directly substitute their values in the equation to get the value of d
d = (64 x 13.53)/1.893 the value of C is gotten from question a
d = 157464/6.75 this is because (64 x 13.53) = 157464 and also 1.893 = 6.75 rounded to two decimal places.
d = 23326.62 rounded to two decimal places
For part c, the equation has three variables, C, b and d. therefore, there will be three variables and hence this function essentially represents a family of functions based on varying values for the boats beam, b. For instance, if the beams of 6, 8 and 10 feet are considered, then the equation will give three functions for each value of b.
C = 4d-1/3 x 6 = 24d-1/3
C = 4d-1/3 x 8 = 32d-1/3
C = 4d-1/3 x 10 = 40d-1/3
The graphs of the functions are shown.
All of the functions are safe for the sailing in the sea since the value of C is less than 2. It is noted that a larger beam can carry a larger displacement and still remain safe.
In conclusion, radical formulas can answer many equations used in real life situations. Using the same rules for solving radicals, then manipulating these equations becomes easy. This technique has helped in determining safety factors of various real life situations.
Dugopolski, M. (2012). Elementary and intermediate algebra (4th ed.). New York, NY:
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