Question

examples:

1. graph points a (3, 7), b (3, 4), and c (9, 4) and the segments connecting them. find the distance from a to b and b to c. find the distance from a to c using both the distance formula and the pythagorean theorem.
2. a school flag measures 3 feet by 5 feet. find the amount of fabric needed to make seven flags, and the amount of binding needed to go around all the edges of all the flags. (sketch a picture and label the information given.)
3. a standard piece of printer paper measures about 38 cm from top to bottom and about 22 cm from side to side. if margins on all sides are about 2.5 cm, how much space is available for the content on the page? (sketch a picture and label the information given.)
4. the gomez family is redoing their family room. it measures 20 ft by 25 ft by 10 ft high. they are covering the floor with square tiles that measure 1 ft on a side and cost $1 each. they also want to add crown molding around the top of all the walls. the molding costs $10 for a 6.5 ft strip. finally, they are going to repaint. one gallon of the paint they chose costs $30 and covers about 400 sq ft. how much is each job going to cost? (sketch a picture and label the information given.)
5. a company ships wooden crates to other countries. they have two different sizes of crates. one is 8 ft by 12 ft by 3 ft. the other is 8 ft by 10 ft by 5 ft. which one requires more wood to build? which one has more space inside? (sketch a picture and label the information given.)

Explanation:

1.

Step1: Find distance from A to B

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For points A$(3,7)$ and B$(3,4)$, $x_1 = 3,y_1=7,x_2 = 3,y_2 = 4$. Then $d_{AB}=\sqrt{(3 - 3)^2+(4 - 7)^2}=\sqrt{0+( - 3)^2}=3$.

Step2: Find distance from B to C

For points B$(3,4)$ and C$(9,4)$, $x_1 = 3,y_1 = 4,x_2=9,y_2 = 4$. Then $d_{BC}=\sqrt{(9 - 3)^2+(4 - 4)^2}=\sqrt{6^2+0}=6$.

Step3: Find distance from A to C using distance formula

For points A$(3,7)$ and C$(9,4)$, $x_1 = 3,y_1 = 7,x_2=9,y_2 = 4$. Then $d_{AC}=\sqrt{(9 - 3)^2+(4 - 7)^2}=\sqrt{6^2+( - 3)^2}=\sqrt{36 + 9}=\sqrt{45}=3\sqrt{5}$.

Step4: Find distance from A to C using Pythagorean theorem

Since $AB = 3$ and $BC=6$, and $\triangle ABC$ is a right - triangle (AB is vertical and BC is horizontal), by the Pythagorean theorem $AC^2=AB^2+BC^2$. So $AC=\sqrt{3^2+6^2}=\sqrt{9 + 36}=\sqrt{45}=3\sqrt{5}$.

Step1: Find area of one flag

The area of a rectangle is $A = l\times w$. For a flag with $l = 5$ feet and $w = 3$ feet, $A_1=3\times5 = 15$ square feet.

Step2: Find area of seven flags

$A_{total}=7\times A_1=7\times15 = 105$ square feet.

Step3: Find perimeter of one flag

The perimeter of a rectangle is $P=2(l + w)$. For a flag, $P_1=2(3 + 5)=16$ feet.

Step4: Find perimeter of seven flags

$P_{total}=7\times P_1=7\times16 = 112$ feet.

Step1: Find available length

The available length of the paper for content is $l=38-(2\times2.5)=38 - 5=33$ cm.

Step2: Find available width

The available width of the paper for content is $w=22-(2\times2.5)=22 - 5 = 17$ cm.

Step3: Find available area

The available area $A=l\times w=33\times17 = 561$ square cm.

Answer:

$d_{AB}=3$, $d_{BC}=6$, $d_{AC}=3\sqrt{5}$

2.