Carpenters' Table Building Efficiency

GeoGebra Commands & Parameters

1. LaTeX(steps) → counter
2. LaTeX("\text{Move the slider to solve the problem.}") → text0
3. Corner(1) → A
4. Corner(3) → B
5. LaTeX("\text{Define the variables in the function.}") → text1
6. LaTeX("\text{Graph the simplified equation.}") → text6
7. Max(list1) → coords
8. LaTeX("\textcolor{#000099}{x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}") → text13
9. LaTeX((((("\text{Let }\textcolor{#000099}{a=1}\text{, }\textcolor{#000099}{b=–" * linterm) * "}\text{, }\textcolor{#000099}{c=–") * prod) * "}\text{, and }\textcolor{#000099}{x=t}.")) → text14
10. LaTeX("\text{Simplify and divide.}") → text16
11. LaTeX("\text{Use the buttons below to return to the original problem or}\cr\text{randomly generate a similar practice problem.}") → RandomizationText
12. LaTeX("\text{Substitute the defined variables into the given function and simplify.}") → text3
13. LaTeX((("\text{Using the given information, let }\textcolor{#000099}{t_1 = t}\text{ hours, }\cr\textcolor{#000099}{t_2 = t + " * a) * "}\text{ hours, and }\textcolor{#000099}{t_{both} = 4}\text{ hours.}")) → text2
14. LaTeX((((("\text{The simplified function can be written as }\textcolor{#000099}{0=t^2-" * linterm) * "t-") * prod) * "}.")) → text5
15. LaTeX((((("\text{The equation can be written as }\textcolor{#000099}{f(t) = t^2-" * linterm) * "t-") * prod) * "}.")) → text7
16. LaTeX("\text{Describe the domain of the function and what that implies for the number }\cr\text{of possible solutions for }\textcolor{#000099}{t}.") → text8
17. LaTeX("\text{Since }\textcolor{#000099}{t}\text{ represents time, the domain of }\textcolor{#000099}{t}\text{ is values of }\textcolor{#000099}{t}\text{ that are greater than }\textcolor{#000099}{0}\text{. }\cr\text{Therefore, only the t-intercept on the positive t-axis (the horizontal axis) gives }\cr\text{a real-world solution for }\textcolor{#000099}{t}.") → text9
18. LaTeX((((("\text{Solve }\textcolor{#000099}{0 = t^2-(" * linterm) * ")t-(") * prod) * ")}\text{ to find the indicated domain value(s) of }\textcolor{#000099}{t}\text{ and determine }\cr\text{the amount of time Benjamin and Ava each take to build a table on their own.}")) → text10
19. LaTeX("\text{To determine the time required for each carpenter to build the table when }\cr\text{working separately, use the quadratic formula.}") → text11
20. LaTeX("\text{Quadratic formula}") → text12
21. LaTeX((((((((((("\text{Only the positive value, }\textcolor{#000099}{t = " * s1) * " + ") * s2) * "}\text{, will result in a positive value of }\textcolor{#000099}{t} \text{;}\cr \text{therefore, }\textcolor{#000099}{t=") * s1) * "+ ") * s2) * " \approx ") * ben) * "}\text{ hours.}")) → text18
22. LaTeX((((((("\text{The variable }\textcolor{#000099}{t}\text{ represents Benjamin’s time, so he can build the table by himself in }\cr\text{approximately } \textcolor{#000099}{" * ben) * "\text{ hours}}\text{. Since }\textcolor{#000099}{t + 2}\text{ represents Ava’s time substitute }\textcolor{#000099}{") * ben) * "}\cr\text{for }\textcolor{#000099}{t}\text{ and solve to determine that Ava can build the table by herself in}\cr\text{approximately }\textcolor{#000099}{") * ava) * "\text{ hours}}.")) → text19
23. LaTeX((((("\textcolor{#000099}{t_1=t}\text{ hours}\cr\textcolor{#000099}{t_2=t+" * a) * "}\text{ hours}\cr\textcolor{#000099}{t_{both}=") * t_{both}) * "} \text{ hours}")) → Problem
24. TableText({{LaTeX["\textcolor{#000099}{t_{both}=\frac{t_1 \cdot t_2}{t_1+t_2}}"], LaTeX["\text{Given function}"]}, {LaTeX[(((((("\textcolor{#000099}{(" * t_{both}) * ")=\frac{(t) \cdot (t +") * a) * ")}{(t) + (t +") * a) * ")}}")], LaTeX[(((("\text{Substitute }\textcolor{#000099}{t}\text{ for }\textcolor{#000099}{t_1}\text{, }\textcolor{#000099}{t+" * a) * "}\text{ for }\textcolor{#000099}{t_2}\text{, and }\textcolor{#000099}{") * t_{both}) * "}\text{ for }\textcolor{#000099}{t_{both}}.")]}, {LaTeX[(((((("\textcolor{#000099}{" * t_{both}) * "=\frac{t^2+") * a) * "t}{2t+") * a) * "}}")], LaTeX["\text{Simplify the numerator and denominator.}"]}, {LaTeX[(((((("\textcolor{#000099}{" * t_{both}) * "(2t+") * a) * ")=t^2+") * a) * "t}")], LaTeX[(("\text{Multiply both sides by }\textcolor{#000099}{2t+" * a) * "}.")]}, {LaTeX[(((((("\textcolor{#000099}{" * dble) * "t+") * prod) * "=t^2+") * a) * "t}")], LaTeX["\text{Distribute.}"]}, {LaTeX[(((("\textcolor{#000099}{0=t^2-" * linterm) * "t-") * prod) * "}")], LaTeX[(((("\text{Subtract }\textcolor{#000099}{" * dble) * "t+") * prod) * "}\text{ from both sides.}")]}}) → text4
25. LaTeX((((((("\textcolor{#000099}{(t)=\frac{-(-" * linterm) * ")\pm\sqrt{(-") * linterm) * ")^2-4(1)(-") * prod) * ")}}{2(1)}}")) → text15
26. LaTeX((((("\textcolor{#000099}{t=" * s1) * "\pm") * s2) * "}")) → text17