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Sagot :
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Michelle is measuring the dimensions of her living room so she can install new flooring. She measures the length at [tex]4\sqrt{12}[/tex] fett with her radical measure tape, and then measures the width at [tex]3\sqrt{8}[/tex] feet.
1. Determine the area of the living room. Leave your answer as a simplified radical expression and include units.
2. Michelle decided she only wants to cover [tex]\frac{3}{4}[/tex] of the room in the new flooring. Determine the area of the flooring she will need to cover [tex]\frac{3}{4}[/tex] of the room. Leave your answer as a simplified radical expression and include units.
3. For the remaining portion of the living room not covered by the new flooring, Michelle wants to paint the concrete. After carefully measuring the exposed concrete, she determines 6 square feet are already painted and the remaining unpainted area forms a square with side length x. The entire concrete, including the painted and unpainted portions, is a total of 31 square feet. write an expression that could be used to find the unpainted square side length x.
4. After installing the new flooring, Michelle palns to install new baseboards around the perimeter of the room. Help Michelle by determining the perimeter of the room, leaving your answer as a simplified radical expression.
5. When Michelle goes to the local hardware store to purchase the baseboards, the shop owner tell her his conversion calculator is broken and he only has 14 yards of baseboard in stock. Help Michelle by converting the perimeter to yards and determine if the hardware store has enough baseboard for her room. Hint: 3 feet are 1 yard.
Answer: 1. [tex]48\sqrt{6}[/tex] ft²
2. [tex]36\sqrt{6}[/tex] ft²
3. [tex]x^{2}+6=31[/tex]
4. [tex](16\sqrt{3}+12\sqrt{2})[/tex] ft
5. No, because perimeter is 15 yards.
Step-by-step explanation:
1. Michelle's living room is a rectangle. Area of a rectangle is calculated as
A = length*width
[tex]A=(4\sqrt{12})(3\sqrt{8})[/tex]
[tex]A=12\sqrt{96}[/tex]
Square root of 96 is not an exact number, so we can only simplify it:
[tex]A=12\sqrt{2^{4}.6}[/tex]
[tex]A=12.4\sqrt{6}[/tex]
[tex]A=48\sqrt{6}[/tex]
Living room has area of [tex]48\sqrt{6}[/tex] square feet.
2. If only 3/4 of the room will be floored, she will need:
[tex]A_{1}=\frac{3}{4}.48\sqrt{6}[/tex]
[tex]A_{1}=36\sqrt{6}[/tex]
Michelle will need [tex]36\sqrt{6}[/tex] square feet of flooring.
3. According to her measure, 6 square feet is painted and a square of size x is unpainted. Both the total of both area is 31.
Area of a square is given by
A = side²
Area of the unpainted square :
[tex]A_{s}=x^{2}[/tex]
Total area is the sum of unpainted and painted area. Then:
[tex]x^{2}+6=31[/tex]
The equation used to determine the unpainted area is [tex]x^{2}+6=31[/tex]
4. Perimeter is the sum of all the sides of a polygon. Since rectangle has two parallel, equal and opposite sides, perimeter is
P = 2*length + 2*width
[tex]P=2(4\sqrt{12})+2(3\sqrt{8})[/tex]
[tex]P=8\sqrt{12}+6\sqrt{8}[/tex]
Simplifying, we have:
[tex]P=8\sqrt{2^{2}.3}+6\sqrt{2^{2}.2}[/tex]
[tex]P=16\sqrt{3}+12\sqrt{2}[/tex]
Perimeter of the room is [tex](16\sqrt{3}+12\sqrt{2})[/tex] feet.
5. Converting feet to yard
3 ft --------------------- 1 yard
[tex](16\sqrt{3}+12\sqrt{2})[/tex] ---------- x
[tex]x=\frac{16\sqrt{3}+12\sqrt{2} }{3}[/tex]
[tex]x=\frac{16\sqrt{3} }{3} +4\sqrt{2}[/tex]
Knowing that [tex]\sqrt{3} =1.73[/tex] and [tex]\sqrt{2}=1.41[/tex], x is approximately
x ≈ 15 yards
So, the hardware store won't have enough baseboard for the room.
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