Question

1.02: swbat solve perimeter and area problems using linear and quadratic equations (day 1)
show all work and annotations for full credit.
#1.) find the area and perimeter of the figures below. include units!
#2.) the rectangle given is one wall of jordans 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) ft.², rebekah substituted in 3 and luis used x = -7. which students substitution do you agree with? what would you tell the other student so they dont make the same mistake?

Explanation:

Step1: Calculate 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, so $P=4 + 5+\sqrt{4^{2}+5^{2}}=9+\sqrt{16 + 25}=9+\sqrt{41}\approx9 + 6.40=15.40$ feet.

Step2: Calculate area and perimeter of second - 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 formula for a rectangle is $P = 2(l + w)$, so $P=2(7 + 2)=18$ feet.

Step3: Calculate area and perimeter of third - 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 formula for a rectangle is $P=2(l + w)$, so $P=2(7+3)=20$ feet.

Step4: Solve for $x$ in the second problem

For a rectangle, opposite sides are equal. So $RU=TS$ and $UT = RS$. But we can also use the perimeter formula $P=2(l + w)$. Given $P = 48$ feet, and the sides $RU = 6x$, $UT=4x + 4$, $TS=5x + 2$, $RS=x + 10$. Since $RU=TS$ and $UT = RS$ (rectangle properties), we have $6x=5x + 2$ (which gives $x = 2$). The sides of the rectangle are: $RU=6x=12$ feet, $UT=4x + 4=12$ feet, $TS=12$ feet, $RS=12$ feet. The sum of the three sides $SR+RU + UT=(x + 10)+6x+(4x + 4)=11x+14$. Substituting $x = 2$, we get $11\times2+14=22 + 14=36$ feet. The cost of the border is $36\times2.99=\$107.64$.

Step5: Analyze the third problem

The area of a figure cannot be negative. The area expression is $A=(8 - 11x)$ square - feet. If $x=-7$, then $A=8-11\times(-7)=8 + 77 = 85$ square feet. If $x = 3$, then $A=8-11\times3=8 - 33=-25$ square feet. Since area cannot be negative, Luis is correct. We would tell Rebekah that in the context of area (a non - negative quantity), we must choose the value of $x$ that results in a non - negative area value.

Answer:

• Triangle: Area = 10 square feet, Perimeter $\approx15.40$ feet
• Second rectangle: Area = 14 square feet, Perimeter = 18 feet
• Third rectangle: Area = 21 square feet, Perimeter = 20 feet
• Second problem: Cost of border = $\$107.64$
• Third problem: Luis is correct. We should choose the value of $x$ that gives a non - negative area value.