Question

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#1.) find the area and perimeter of the figures below. include units!
**#2.) the rectangle given is one wall of jordan’s bedroom. ru = 6x feet, ut = 4x + 4 feet, ts = 5x + 2 feet, and rs = x + 10 feet and the perimeter is 48 feet. jordan is putting a border around sr, ru, and ut. the border costs $2.99 a foot. how much money will it cost jordan to put a border around those three sides?
**#3.) when solving for the area of a triangle, rebekah and luis had to solve a quadratic equation. both students got x = -7 and x = 3 for their solutions. when it came to finding the area given the expression (8 - 11x) ft², rebekah substituted in 3 and luis used x = -7. which student’s substitution do you agree with? what would you tell the other student so they don’t make the same mistake?

Explanation:

Step1: Find area and perimeter of first - triangle

The area formula for a triangle is $A=\frac{1}{2}bh$, where $b = 5$ feet and $h=4$ feet. So $A=\frac{1}{2}\times5\times4=10$ square feet. The perimeter $P$ of a triangle is the sum of its sides, $P=4 + 5+\sqrt{4^{2}+5^{2}}=9+\sqrt{16 + 25}=9+\sqrt{41}\approx9 + 6.4=15.4$ feet.

Step2: Find area and perimeter of first - rectangle

The area formula for a rectangle is $A = lw$, where $l = 7$ feet and $w = 2$ feet. So $A=7\times2 = 14$ square feet. The perimeter $P=2(l + w)=2(7 + 2)=18$ feet.

Step3: Find area and perimeter of second - rectangle

The area formula for a rectangle is $A=lw$, where $l = 7$ feet and $w = 3$ feet. So $A=7\times3=21$ square feet. The perimeter $P=2(l + w)=2(7 + 3)=20$ feet.

Step4: Solve for $x$ in the second - problem

For a rectangle, the perimeter formula is $P = 2(l + w)$. Here, $P=48$ feet. Let $RU = 6x$, $UT=4x + 4$, $TS=5x+2$, $RS=x + 10$. Since opposite sides of a rectangle are equal, $RU=TS$ and $UT=RS$. But using the perimeter formula $P=2(RU + UT)$. So $48=2(6x+(4x + 4))$. First, simplify the equation: $48=2(10x + 4)=20x+8$. Subtract 8 from both sides: $48-8=20x$, so $40 = 20x$. Then $x = 2$. The lengths of the sides are $RU=6x=12$ feet, $UT=4x + 4=12$ feet, $RS=x + 10=12$ feet, $TS=5x+2=12$ feet. The length of the three - side border is $SR+RU+UT=12 + 12+12=36$ feet. The cost of the border is $36\times2.99=\$107.64$.

Step5: Analyze the quadratic - solution in the third - problem

The area of a triangle is given by a non - negative value. The area expression is $(8 - 11x)ft^{2}$. If $x=-7$, then $8-11x=8+77 = 85$. If $x = 3$, then $8-11x=8-33=-25$. Since the area of a triangle cannot be negative, Rebekah's substitution of $x = 3$ is incorrect and Luis's substitution of $x=-7$ is correct. I would tell Rebekah that the area of a geometric shape (in this case, a triangle) must be non - negative, so she should choose the value of $x$ that makes the area expression non - negative.

Answer:

1.

• Triangle: Area = 10 square feet, Perimeter $\approx15.4$ feet
• First rectangle: Area = 14 square feet, Perimeter = 18 feet
• Second rectangle: Area = 21 square feet, Perimeter = 20 feet

1. Cost of border = $\$107.64$
2. I agree with Luis's substitution. I would tell Rebekah that the area of a triangle must be non - negative, so she should choose the value of $x$ that makes the area expression $(8 - 11x)$ non - negative.