## WeBWorK Problems

### Formula with units not recognizing correct answer.

by tim Payer -
Number of replies: 0
Greetings,

I am having a small problem with a FormulaWithUnits() application.
The correct answer is "1/A hr" but Webwork replies that:

The Variable "A" has been declared as a Real variable, and the second answer in this assignment has "A/e hr" as a correct answer and this problem as no such problems as the "1/A hr" answer does.

The only temporary work around is to ask the student to enter parentheses around the formula: "(1/A) hr". And with this webwork will function correctly in returning a green field for a correct answer for the student.

The code is below and the formulas are declared on lines 20 and 21.

Sincerely, Tim

# Webwork Workshop for Payer, Homework 1, Problem 1:
# Given a function for the probability of mortality at an age x,
# Find the the age that has the highest probability for death.
# So the student should know to take the derivative of the function,
# set the derivative to zero, and then solve for x.
DOCUMENT();
"MathObjects.pl",
"PGML.pl",
"parserNumberWithUnits.pl",
"parserFormulaWithUnits.pl");

Context("Numeric");
$a = Real(random(3,13,2));$b = Real(random(2,9,1));
$c = Real(random(2,10,2));$d = Real(random(2,10,1));
$ans1 = Compute("1/A");$ans11 = FormulaWithUnits("$ans1 hr");$ans2 = Compute("A/e");
$ans22 = FormulaWithUnits("$ans2 hr");

TEXT(beginproblem());
BEGIN_PGML
A biologist measures the probability of mortality for a given species of mayfly by the function:
[P(x) = A^2xe^{-Ax}]
where [x] is the adult mayfly's age in hours such that [0 < A < e].

Find the life-span age that corresponds with the maximum probability for a mayfly's death. Your answer should be in terms of A and you will have to *wrap the expression in parentheses*.

[_____]{$ans11}{20} What age has the maximum probability for mayfly mortality? Again your answer should be in terms of A. [________]{$ans22}{20}

END_PGML

BEGIN_PGML_SOLUTION
*SOLUTION*

1.) We must take the derivative and set the derivative to zero to solve for x, the age for the most likely mayfly life span.

[P(x) = A^2xe^{-Ax}]

[P(x) = A^2(xe^{-Ax})] First pull out the constant of [A^2]

[::P'(x) = A^2(x'e^{-Ax}+xe^{-Ax}'{-Ax}')::] Apply the prime tics for the product rule and chain rule.

[P'(x) = A^2(e^{-Ax} + xe^{-Ax}(-A))] Take the derivative.

[P'(x) = A^2e^{-Ax}(1-Ax)] Pull the common factor and reduce.

[A^2e^{-Ax}(1-Ax) = 0] Set the derivative to zero and solve for x to find the critical point.

[1-Ax = 0] Recognize that [A^2] and [e^{-Ax}] can not be zero as both are positive.

[Ax = 1]

[x = \frac{1}{A}] Which is the age in hours for the life span with the greatest chance of death for the mayfly.

2.) Evaluate P(x) at the critical point of [x = \frac{1}{A}] for the mayfly's age that has the maximum probability for death.

[P(x) = A^2xe^{-Ax}]

[P(\frac{1}{A}) = A^2\frac{1}{A}e^{-A\frac{1}{A}}]

[P(\frac{1}{A}) = Ae^{-1} = \frac{A}{e}]

END_PGML_SOLUTION

ENDDOCUMENT();