Question

simplify (z-\frac{z}{3}) if (y = 4) and (z = 3). you are responsible for buying the hamburger rolls for an upcoming picnic. each bag of rolls costs $1.30 and contains 8 rolls. you need to buy a total of 64 rolls. how much money will it cost for the rolls? a teacher wants to buy note - pads for the entire class. if note pads cost $1.75 each, write an equation that shows how many can be bought with $40. what is the scale factor from (\triangle tuv) to (\triangle qrs)? if (\triangle1) is the pre - image, which triangles could be the image after a reflection? steven has a rectangular closet. the length is 10 feet and the width is 48 inches. determine the area of the closet in square feet. solve the equation for (x): (\frac{x + 4}{x+9}=\frac{6}{5}), (\frac{x}{2}-8 = 17), (4(8x - 8)=-192), (2(2x - 1)=x). draw a new figure that has been dilated by a factor of (\frac{1}{2}) (centered at the origin).

Explanation:

1. Area of the closet:

• First, convert the width from inches to feet. Since 1 foot = 12 inches, if the width is 48 inches, then the width in feet is $\frac{48}{12}=4$ feet.
• The area of a rectangle is given by $A = l\times w$, where $l$ is the length and $w$ is the width. Here, $l = 10$ feet and $w = 4$ feet.
• So, $A=10\times4 = 40$ square - feet.

1. Cost of hamburger rolls:

• Each bag contains 8 rolls. You need 64 rolls. The number of bags, $n=\frac{64}{8}=8$ bags.
• Each bag costs $1.30$. The total cost, $C = 1.30\times8=10.40$ dollars.

1. Equation for note - pads:

• Let the number of note - pads be $n$. Each note - pad costs $1.75$, and you have $40$ dollars. The equation is $1.75n = 40$.

1. Solving $\frac{x + 4}{x + 9}=\frac{6}{5}$:

• Cross - multiply: $5(x + 4)=6(x + 9)$.
• Expand: $5x+20 = 6x + 54$.
• Subtract $5x$ from both sides: $20=x + 54$.
• Subtract 54 from both sides: $x=20 - 54=-34$.

1. Solving $\frac{x}{2}-8 = 17$:

• Add 8 to both sides: $\frac{x}{2}=17 + 8=25$.
• Multiply both sides by 2: $x = 25\times2=50$.

1. Solving $4(8x - 8)=-192$:

• First, divide both sides by 4: $8x-8=\frac{-192}{4}=-48$.
• Add 8 to both sides: $8x=-48 + 8=-40$.
• Divide both sides by 8: $x=\frac{-40}{8}=-5$.

1. Solving $2(2x - 1)=x$:

• Expand the left - hand side: $4x-2=x$.
• Subtract $x$ from both sides: $4x-x-2 = 0$, so $3x-2 = 0$.
• Add 2 to both sides: $3x=2$.
• Divide both sides by 3: $x=\frac{2}{3}$.

1. Scale factor from $\triangle TUV$ to $\triangle QRS$:

• To find the scale factor, we can compare the lengths of corresponding sides. For example, if we consider the vertical sides, the length of the vertical side of $\triangle TUV$ (from $U(0,3)$ to $V(1,1)$) is $3 - 1 = 2$, and the length of the vertical side of $\triangle QRS$ (from $R(0,6)$ to $S(2,2)$) is $6 - 2 = 4$.
• The scale factor $k=\frac{\text{length of side in }\triangle QRS}{\text{length of corresponding side in }\triangle TUV}=\frac{4}{2}=2$.

Answer:

1. Area of the closet: 40 square feet
2. Cost of rolls: $10.40$
3. Equation for note - pads: $1.75n = 40$
4. $x=-34$
5. $x = 50$
6. $x=-5$
7. $x=\frac{2}{3}$
8. Scale factor: 2