DEV ProjectThe Avengers

By: Emily White and Ally Bauer

Problem 1Iron Man has 3000 square feet of building material, and he wants to make an extension on Stark Tower with 9 rooms inside. What dimensions would give the maximum area(2 dimensionally)? 3000= 4x+4y A=xy-4y -4y A=(750-y)y3000-4y= 4x -y^2+750y 4 4 -b = -750 = 375 750-y=x 2(a) -2

(750-375)= 375x= 375 y= 375

ExplanationSo he starts out with 3000 feet of building material, and he wants 9 separate rooms. There would be 4 “x” dimensions, and 4 “y” dimensions. The 3000 is the perimeter, so he would solve for 4x+4y. The he would plug in 750-y for x in the area equation. Then he would distribute to make a standard form quadratic equation. He would use the opposite b over 2a equation to find the x of the vertex, then plug in the answer to find the y. Those would give the x and y dimensions that would make the maximum area, which is 140625 square feet.

Problem 2 3y+8=x Find the inverse√5y+2 3y+8^2=x^2 √5y+2 3y+8 =x^2(5y+2)5y+2* 5y+23y+8= 5x^2y+2x^2 -8 -83y-(5x^2y)= 2x^2-8y(3-5x^2)= 2x^2-8 (3-5x^2) (3-5x^2) 2x^2-8y= 3-5x^2

Explanation

To find the inverse, first we had to square both sides. Then we multiplied by 5x +2, and distributed the x^2. Then we subtracted 8, and subtracted 5x^2y so that all of the “y”s would be on the same side. Then we factored out the y, and divided the 3-5x^2. That way, we got the inverse.

Problem 3Factor completely: x^4-2x^3+27x-54(x^4-2x^3)+(27x-54)x^3(x-2)+27(x-2)(x^3+27)(x-2)A^3=x^3 3√A^3= 3√x^3B^3=27 3√B^3= 3√27(x-2)(x+3)(x^2-3x+9)

ExplanationIn order to factor by grouping, we first split the equation into two different quantities. Then we factored out the greatest common factor of each quantity. Was was left between each set of parentheses was the same, so we put them together, and put the x^3 and the 27 together into one quantity. After that we identified the A^3 and B^3 values in “the sum of cubes” equation. We then used “the sum of cubes” format to factor it completely into the final equation.

Problem 4Find the Domain: √x^2-25 √x^2+6x+8x^2-25 ≥0x= +/-5D:(-∞,-5 U 5,∞)x^2+6x+8 ≥0 -5 -4 -2 5

(x+2)(x+4) ≥0 x= -2 x= -4D:(-∞,-4)U(-2,∞)

DF: (-∞,-5 U 5,∞)

ExplanationTo find the domain of a quadratic under a radical, the first thing we did was take them out from under the radical and set each of them equal to 0 (greater than or equal to for the top, equal to for the bottom). Then we factored them, although we only showed it for the one. Then we used the x-intercepts to find the domain of each function separately. Next we found where they overlap, and that was the final domain.

Problem 5Solve by completing the square: 7x^2-7x-18= 07x^2-7x-18= 0 +18 +187(x^2-x+¼)= 79/47(x-½)^2= 19.75 7 7 √(x-½)^2= 19.75

√ 7

x-½= +/- 19.75 +½

+½ √ 7

x= +/- 19.75 +½ √ 7

Explanation

To solve by completing the square, the first thing we had to do was add the c value on both sides. Then we factored out the 7, and used the (b/2)^2 equation to make a new c value. Next we divided both sides by 7, and factored the equation. We then squared both sides and subtracted ½. That gave us the answer.

ReflectionsEmily: Ally and I chose the concepts that we did because we wanted to put our learning to the test, and challenge what we are capable of. The problems that we created provide an overview of our mathematical understanding because they show that we are able to use our knowledge wisely. They display our ability to work our way through problems that are by all means, not the easiest of the bunch. This assignment proved to be fairly beneficial because it forced us to work somewhat backwards, and look at equations from a different perspective.Ally: Emily and I made our project using the problems that we did because this was what we felt was the height of our capabilities. I personally struggled with completing the square and finding the inverse, so making up problems like this enhanced my understanding of the units that we covered. Making u math problems was new to both of us, and it was really helpful for studying the past work that we’ve done. We both struggled a bit with finding problems that would be difficult enough to portray our knowledge, but in the end, I feel as though we did a good job of it.