*Show al work1. A label on an iced tea drink mix says to use 2 tbsp of iced tea mix to make one serving (8 oz) of iced tea.a. What is the ratio of drink mix to iced tea? (Hint: 1 oz = 2 tbsp)b. Suppose you want to make a quart (32 oz) of iced tea. How many ounces of drink mix should be used?c. How many tablespoons of iced tea mix should be used to make a quart (32 oz) of iced tea?2. RST has coordinates R(-4, -3), S(0, 5), and T(7, -3).a. Graph RST in a coordinate plane.b. Find the perimeter of RST.c. Find the area of RST.d. The centroid of a triangle is the intersection of its medians. Use the following steps to find the centroid of RST using segment partition.i. Find the midpoint of. Label it N.ii. Find the point, C, that partitionssuch that SC:CN has a ratio of 2:1 .iii. Plot point C in your graph of RST.e. Find and graph the midpoints of,,and. Draw a segment connecting the vertex opposite each midpoint to the midpoint.i. How does the intersection of the segments compare to point C from question 4?3. Brian is renting an apartment. The floor plan of the apartment is shown below.a. What is the total area of the apartment?b. What are the dimensions of the master bedroom?c. Brian wants to buy a dining room table from the local furniture store. He sees a table that is 9 ft long and 3 ft wide. Would it be reasonable to buy this table to put in the dining room? Why or why not?

Introduction:

The following questions explore different mathematical concepts such as ratios, coordinates, and area calculations. Whether you’re preparing a refreshing drink using a drink mix or trying to determine the dimensions of a room, these problems require a mix of computations and critical thinking.

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Description:

The three-part set of problems begins with calculating the ratio of drink mix to iced tea in order to prepare a certain amount of this beverage. Next, the problem moves to graphing and measuring the perimeter and area of the triangle with the given coordinates. Finally, we analyze the floor plan of an apartment and determine if a specific dining room table would be a suitable fit. Whether you’re interested in refreshing your math skills or testing your knowledge, these questions provide an opportunity to apply mathematical concepts to real-world situations.

Objectives:

– To solve ratio and proportion problems using unit conversion

– To apply coordinate geometry concepts in determining the properties of a triangle

– To calculate area and perimeter of two-dimensional shapes

– To interpret floor plans and make reasonable decisions based on measurements

Learning Outcomes:

By the end of this activity, the students will be able to:

– Calculate the ratio of iced tea drink mix to iced tea for a given recipe

– Convert units of measurement to determine the appropriate amount of ingredient needed

– Graph a triangle given its coordinates and calculate its perimeter

– Use segment partition to find the centroid of a triangle

– Calculate the area of a triangle given its side lengths

– Interpret a floor plan and calculate the area of individual rooms

– Make reasonable decisions based on measurements to determine the suitability of a furniture item for a specific room.

1. Ratio and Proportion

a. Objective: To solve ratio and proportion problems using unit conversion

b. Learning Outcomes:

– Calculate the ratio of drink mix to iced tea for a given recipe

– Convert units of measurement to determine the appropriate amount of drink mix needed to make a quart of iced tea

– Convert units of measurement to determine the number of tablespoons of iced tea mix required to make a quart of iced tea.

2. Coordinate Geometry

a. Objective: To apply coordinate geometry concepts in determining the properties of a triangle

b. Learning Outcomes:

– Graph a triangle given its coordinates

– Calculate the perimeter of a triangle given its vertices

– Calculate the area of a triangle given its side lengths

– Use segment partition to find the centroid of a triangle

– Plot points on a graph and connect them with lines to determine the midpoints of the sides of a triangle.

3. Two-Dimensional Shapes

a. Objective: To calculate area and perimeter of two-dimensional shapes

b. Learning Outcomes:

– Calculate the area of a shape given its dimensions

– Calculate the perimeter of a shape given its dimensions.

4. Floor Plans

a. Objective: To interpret floor plans and make reasonable decisions based on measurements

b. Learning Outcomes:

– Interpret a floor plan and label the dimensions of each room

– Calculate the area of a room given its dimensions

– Make reasonable decisions based on measurements to determine the suitability of a furniture item for a specific room.

Solution 1:

a. The ratio of drink mix to iced tea is 2 tbsp : 8 oz or 1:4 (since 1 oz = 2 tbsp)

b. To make a quart (32 oz) of iced tea, we need 1/4 of the amount of drink mix. So, 32 oz ÷ 4 = 8 oz of drink mix should be used.

c. To make a quart (32 oz) of iced tea, we need 8 oz of drink mix. Since 1 oz = 2 tbsp, we need 8 oz x 2 tbsp/oz = 16 tbsp of iced tea mix to make a quart of iced tea.

Solution 2:

a. Graph RST on a coordinate plane:

b. To find the perimeter of RST, we need to find the length of each side. Using the distance formula:

– To find RS: d = sqrt((0-(-4))^2 + (5-(-3))^2) = sqrt(64 + 64) = 8sqrt(2)

– To find ST: d = sqrt((7-0)^2 + (-3-5)^2) = sqrt(49 + 64) = 13

– To find RT: d = sqrt((7-(-4))^2 + (-3-(-3))^2) = sqrt(121) = 11

The perimeter of RST = RS + ST + RT = 8sqrt(2) + 13 + 11 = 24.313

c. To find the area of RST, we can use the formula for the area of a triangle:

Area = 1/2 * base * height

We can use RT as the base and the height can be found by dropping a perpendicular line from S to RT. To find the height, we need to find the equation of the line passing through R and T, which is y = -6x/11 – 3. Using this equation, we find the coordinate of the point where the perpendicular line intersects RT is (7/2, 1/2). The height is the distance between S and this point, which can be found using the distance formula:

height = d = sqrt((7/2-0)^2 + (1/2-5)^2) = sqrt(7/2^2 + (-9/2)^2) = sqrt(85)/2

Therefore, the area of RST = 1/2 * 11 * sqrt(85)/2 = 28.625

d. To find the centroid of RST using segment partition:

i. The midpoint of RT is ((-4+7)/2, (-3-3)/2) = (3/2, -3)

ii. To find C, we partition SC into two segments with ratio 2:1. This means that SC = 2CN. Let’s call the coordinates of C (x,y) and the coordinates of N (3/2, -3). We can write two equations based on the fact that C is on the line passing through R and S and that SC = 2CN:

y = (8/4)x – 3

d = 2d’

sqrt((x-0)^2 + (y-5)^2) = 2 * sqrt((x-3/2)^2 + (y+3)^2)

Solving for x and y, we get C = (21/5, 3/5).

iii. We can plot point C in the graph of RST as shown below:

e. To find the midpoints of the sides, we can use the midpoint formula:

– The midpoint of RS is ((-4+0)/2, (-3+5)/2) = (-2, 1)

– The midpoint of ST is ((0+7)/2, (5-3)/2) = (7/2, 1)

– The midpoint of RT is ((-4+7)/2, (-3-3)/2) = (3/2, -3)

We can then draw a segment connecting each vertex to its opposite midpoint as shown below:

i. The intersection of the segments is point C, which is the centroid of RST.

Suggested Resources/Books:

1. “Basic Math and Pre-Algebra Workbook For Dummies” by Mark Zegarelli

2. “Geometry: Concepts and Applications” by McGraw Hill Education

3. “Algebra & Geometry: An Introduction to University Mathematics” by Mark Stamp

4. “College Algebra and Trigonometry” by Richard N. Aufmann and Vernon C. Barker

Similar asked questions:

1. How do you convert measurements between ounces and tablespoons in cooking?

2. How do you find the centroid of a triangle using segment partition?

3. How can you calculate the perimeter and area of a triangle given its coordinates?

4. How can you find the midpoints and slopes of the sides of a triangle?

5. How can you determine if a given furniture will fit in a specific room or space?*Show al work1. A label on an iced tea drink mix says to use 2 tbsp of iced tea mix to make one serving (8 oz) of iced tea.a. What is the ratio of drink mix to iced tea? (Hint: 1 oz = 2 tbsp)b. Suppose you want to make a quart (32 oz) of iced tea. How many ounces of drink mix should be used?c. How many tablespoons of iced tea mix should be used to make a quart (32 oz) of iced tea?2. RST has coordinates R(-4, -3), S(0, 5), and T(7, -3).a. Graph RST in a coordinate plane.b. Find the perimeter of RST.c. Find the area of RST.d. The centroid of a triangle is the intersection of its medians. Use the following steps to find the centroid of RST using segment partition.i. Find the midpoint of. Label it N.ii. Find the point, C, that partitionssuch that SC:CN has a ratio of 2:1 .iii. Plot point C in your graph of RST.e. Find and graph the midpoints of,,and. Draw a segment connecting the vertex opposite each midpoint to the midpoint.i. How does the intersection of the segments compare to point C from question 4?3. Brian is renting an apartment. The floor plan of the apartment is shown below.a. What is the total area of the apartment?b. What are the dimensions of the master bedroom?c. Brian wants to buy a dining room table from the local furniture store. He sees a table that is 9 ft long and 3 ft wide. Would it be reasonable to buy this table to put in the dining room? Why or why not?

Introduction:

The following questions explore different mathematical concepts such as ratios, coordinates, and area calculations. Whether you’re preparing a refreshing drink using a drink mix or trying to determine the dimensions of a room, these problems require a mix of computations and critical thinking.

Description:

The three-part set of problems begins with calculating the ratio of drink mix to iced tea in order to prepare a certain amount of this beverage. Next, the problem moves to graphing and measuring the perimeter and area of the triangle with the given coordinates. Finally, we analyze the floor plan of an apartment and determine if a specific dining room table would be a suitable fit. Whether you’re interested in refreshing your math skills or testing your knowledge, these questions provide an opportunity to apply mathematical concepts to real-world situations.

Objectives:

– To solve ratio and proportion problems using unit conversion

– To apply coordinate geometry concepts in determining the properties of a triangle

– To calculate area and perimeter of two-dimensional shapes

– To interpret floor plans and make reasonable decisions based on measurements

Learning Outcomes:

By the end of this activity, the students will be able to:

– Calculate the ratio of iced tea drink mix to iced tea for a given recipe

– Convert units of measurement to determine the appropriate amount of ingredient needed

– Graph a triangle given its coordinates and calculate its perimeter

– Use segment partition to find the centroid of a triangle

– Calculate the area of a triangle given its side lengths

– Interpret a floor plan and calculate the area of individual rooms

– Make reasonable decisions based on measurements to determine the suitability of a furniture item for a specific room.

1. Ratio and Proportion

a. Objective: To solve ratio and proportion problems using unit conversion

b. Learning Outcomes:

– Calculate the ratio of drink mix to iced tea for a given recipe

– Convert units of measurement to determine the appropriate amount of drink mix needed to make a quart of iced tea

– Convert units of measurement to determine the number of tablespoons of iced tea mix required to make a quart of iced tea.

2. Coordinate Geometry

a. Objective: To apply coordinate geometry concepts in determining the properties of a triangle

b. Learning Outcomes:

– Graph a triangle given its coordinates

– Calculate the perimeter of a triangle given its vertices

– Calculate the area of a triangle given its side lengths

– Use segment partition to find the centroid of a triangle

– Plot points on a graph and connect them with lines to determine the midpoints of the sides of a triangle.

3. Two-Dimensional Shapes

a. Objective: To calculate area and perimeter of two-dimensional shapes

b. Learning Outcomes:

– Calculate the area of a shape given its dimensions

– Calculate the perimeter of a shape given its dimensions.

4. Floor Plans

a. Objective: To interpret floor plans and make reasonable decisions based on measurements

b. Learning Outcomes:

– Interpret a floor plan and label the dimensions of each room

– Calculate the area of a room given its dimensions

– Make reasonable decisions based on measurements to determine the suitability of a furniture item for a specific room.

Solution 1:

a. The ratio of drink mix to iced tea is 2 tbsp : 8 oz or 1:4 (since 1 oz = 2 tbsp)

b. To make a quart (32 oz) of iced tea, we need 1/4 of the amount of drink mix. So, 32 oz ÷ 4 = 8 oz of drink mix should be used.

c. To make a quart (32 oz) of iced tea, we need 8 oz of drink mix. Since 1 oz = 2 tbsp, we need 8 oz x 2 tbsp/oz = 16 tbsp of iced tea mix to make a quart of iced tea.

Solution 2:

a. Graph RST on a coordinate plane:

b. To find the perimeter of RST, we need to find the length of each side. Using the distance formula:

– To find RS: d = sqrt((0-(-4))^2 + (5-(-3))^2) = sqrt(64 + 64) = 8sqrt(2)

– To find ST: d = sqrt((7-0)^2 + (-3-5)^2) = sqrt(49 + 64) = 13

– To find RT: d = sqrt((7-(-4))^2 + (-3-(-3))^2) = sqrt(121) = 11

The perimeter of RST = RS + ST + RT = 8sqrt(2) + 13 + 11 = 24.313

c. To find the area of RST, we can use the formula for the area of a triangle:

Area = 1/2 * base * height

We can use RT as the base and the height can be found by dropping a perpendicular line from S to RT. To find the height, we need to find the equation of the line passing through R and T, which is y = -6x/11 – 3. Using this equation, we find the coordinate of the point where the perpendicular line intersects RT is (7/2, 1/2). The height is the distance between S and this point, which can be found using the distance formula:

height = d = sqrt((7/2-0)^2 + (1/2-5)^2) = sqrt(7/2^2 + (-9/2)^2) = sqrt(85)/2

Therefore, the area of RST = 1/2 * 11 * sqrt(85)/2 = 28.625

d. To find the centroid of RST using segment partition:

i. The midpoint of RT is ((-4+7)/2, (-3-3)/2) = (3/2, -3)

ii. To find C, we partition SC into two segments with ratio 2:1. This means that SC = 2CN. Let’s call the coordinates of C (x,y) and the coordinates of N (3/2, -3). We can write two equations based on the fact that C is on the line passing through R and S and that SC = 2CN:

y = (8/4)x – 3

d = 2d’

sqrt((x-0)^2 + (y-5)^2) = 2 * sqrt((x-3/2)^2 + (y+3)^2)

Solving for x and y, we get C = (21/5, 3/5).

iii. We can plot point C in the graph of RST as shown below:

e. To find the midpoints of the sides, we can use the midpoint formula:

– The midpoint of RS is ((-4+0)/2, (-3+5)/2) = (-2, 1)

– The midpoint of ST is ((0+7)/2, (5-3)/2) = (7/2, 1)

– The midpoint of RT is ((-4+7)/2, (-3-3)/2) = (3/2, -3)

We can then draw a segment connecting each vertex to its opposite midpoint as shown below:

i. The intersection of the segments is point C, which is the centroid of RST.

Suggested Resources/Books:

1. “Basic Math and Pre-Algebra Workbook For Dummies” by Mark Zegarelli

2. “Geometry: Concepts and Applications” by McGraw Hill Education

3. “Algebra & Geometry: An Introduction to University Mathematics” by Mark Stamp

4. “College Algebra and Trigonometry” by Richard N. Aufmann and Vernon C. Barker

Similar asked questions:

1. How do you convert measurements between ounces and tablespoons in cooking?

2. How do you find the centroid of a triangle using segment partition?

3. How can you calculate the perimeter and area of a triangle given its coordinates?

4. How can you find the midpoints and slopes of the sides of a triangle?

5. How can you determine if a given furniture will fit in a specific room or space?

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