MATH SOLVE

3 months ago

Q:
# PROMPTEiffel Tower: Francesca wants to glue this 4-inch by 6-inch photo on top of a mat that will increase the length of each side by another x inches. Find an expression representing the area of the mat, and use this expression to find the area and cost of the mat for different values of x.YOUR ASSIGNMENTChoosing a PhotoFrancesca wants to give one of her photographs to a friend as a gift. Help her choose the photograph and determine the style and size of the mat she can afford.1. Describe the design and original dimensions of the photo you chose for Francesca to give as a gift. (2 points: 1 point for describing the photo and 1 point for including the dimensions)Photo description: Dimensions (including units): Finding the Total Area of the Photo and Mat2. Francesca wants to glue her photo on top of a mat that will increase the length of each side. The mat border has a width of x, so the length of each side would increase by 2x inches.Represent the length and width of the mat with an expression. (2 points)Length:Width:3. The area of the mat is the product of the length and width. Write an expression for area as the product of two binomials. Do not multiply. (1 point)Area:4. Find the polynomial expression that represents the area of the mat, using the FOIL method to multiply the two binomials. Show your work. (3 points)Polynomial expression:Determining the Cost of the Mat5. What units would you expect to use for this area? Why? (2 points)6. To verify your polynomial expression, multiply the factors from Question 3 using the distributive property. Show your work below. (2 points)Multiply factors using the distribution method:7. Write your polynomial expression representing the total area of the mat in the top of the right table column below. Then evaluate the expression to find the total areas when the mat extends 1 or 2 inches beyond the picture on each side. Don't forget to add units. (2 points: 1 point for each mat area) x 1.0 in 2.0 in 8. If a white mat costs $0.03 per square inch and a black mat costs $0.05 per square inch, determine the cost of each size of black and white mat. To get started, copy your previous calculations for the total areas of the mats into the first column of the table below. (4 points: 2 points for the costs of the white mats and 2 points for the costs of the black mats) x Total area of mat Cost of white mat Cost of black mat 1.0 in 2.0 in 9. Select the size and color of the mat you think Francesca should use for the photograph. Identify the cost and why you selected this mat.

Accepted Solution

A:

1.
Photo description: A picture of the Eiffel tower, to be stuck on a mat.

Dimensions (including units): 4 in x 6 in

2. Since 2x would be added to each dimension:

Length: 6 + 2x (inches)

Width: 4 + 2x (inches)

3. Area: A = LW = (6+2x)(4+2x) square inches

4. F: (6)(4) = 24, O: (6)(2x) = 12x, I: (2x)(4) = 8x, L: (2x)(2x) = 4x^2

Polynomial expression: Adding the FOIL terms up: 4x^2 + 20x + 24

5. The area should be in square inches, since we multiplied length (in inches) by width (in inches).

6. Multiply factors using the distribution method:

(6+2x)(4+2x) = 6(4+2x) + 2x(4+2x) = 24 + 12x + 8x + 4x^2 = 24 + 20x + 4x^2

This is identical to the expression in Part 4.

7. x: 24 + 20x + 4x^2

If x = 1.0 in: Area = 24 + 20(1) + 4(1)^2 = 48 in^2

If x = 2.0 in: Area = 24 + 20(2) + 4(2)^2 = 80 in^2

8. If a white mat costs $0.03 per square inch and a black mat costs $0.05 per square inch, determine the cost of each size of black and white mat.

x Total area of mat Cost of white mat Cost of black mat

1.0 in, A = 48 in^2, (0.03)(48) = $1.44, (0.05)(48) = $2.40

2.0 in, A = 80 in^2, (0.03)(80) = $2.40, (0.05)(80) = $4.00

9. The cheapest option would be the white mat with 1-in margins on all sides, which would cost $1.44. Without any further criteria on aesthetics or size limitations, this is the most viable option.

Dimensions (including units): 4 in x 6 in

2. Since 2x would be added to each dimension:

Length: 6 + 2x (inches)

Width: 4 + 2x (inches)

3. Area: A = LW = (6+2x)(4+2x) square inches

4. F: (6)(4) = 24, O: (6)(2x) = 12x, I: (2x)(4) = 8x, L: (2x)(2x) = 4x^2

Polynomial expression: Adding the FOIL terms up: 4x^2 + 20x + 24

5. The area should be in square inches, since we multiplied length (in inches) by width (in inches).

6. Multiply factors using the distribution method:

(6+2x)(4+2x) = 6(4+2x) + 2x(4+2x) = 24 + 12x + 8x + 4x^2 = 24 + 20x + 4x^2

This is identical to the expression in Part 4.

7. x: 24 + 20x + 4x^2

If x = 1.0 in: Area = 24 + 20(1) + 4(1)^2 = 48 in^2

If x = 2.0 in: Area = 24 + 20(2) + 4(2)^2 = 80 in^2

8. If a white mat costs $0.03 per square inch and a black mat costs $0.05 per square inch, determine the cost of each size of black and white mat.

x Total area of mat Cost of white mat Cost of black mat

1.0 in, A = 48 in^2, (0.03)(48) = $1.44, (0.05)(48) = $2.40

2.0 in, A = 80 in^2, (0.03)(80) = $2.40, (0.05)(80) = $4.00

9. The cheapest option would be the white mat with 1-in margins on all sides, which would cost $1.44. Without any further criteria on aesthetics or size limitations, this is the most viable option.