Question

mark is designing a robot out of recycled material for art class. the robot’s head and body will be made of cardboard. the head of the robot is a square and the body is a rectangle. the dimensions are shown below.

1. find the area of the robot’s head.
2. find the area of the robot’s body.
3. find the total amount of cardboard mark will use in his creation.

Explanation:

Question 7:

Step1: Identify the shape and formula

The robot's head is a square. The formula for the area of a square is \( A = s \times s \), where \( s \) is the side length. From the diagram, the side length of the square (head) is \( x - 5 \)? Wait, no, looking at the diagram, the head's side is \( x - 5 \)? Wait, the diagram shows the head with side \( x - 5 \)? Wait, no, the vertical side of the head is labeled \( x - 5 \), and since it's a square, both length and width are \( x - 5 \). Wait, no, maybe I misread. Wait, the head is a square, so area is side squared. Wait, the diagram: the head is a square with side \( x - 5 \)? Wait, no, the vertical dimension of the head is \( x - 5 \), so since it's a square, length and width are both \( x - 5 \). So area of square is \( (x - 5)^2 \). Wait, no, maybe the side is \( x - 5 \)? Wait, let's check again. The problem says the head is a square. So area of square is side × side. So if the side is \( x - 5 \), then area is \( (x - 5)(x - 5) \).

Wait, maybe I made a mistake. Wait, the diagram: the head is a square, and the vertical side is \( x - 5 \), so the side length is \( x - 5 \). So area of square is \( s^2 \), where \( s = x - 5 \). So:

Step1: Recall area of square formula

Area of square \( A = s^2 \), where \( s \) is the side length.

Step2: Substitute the side length

Here, \( s = x - 5 \), so \( A = (x - 5)^2 \).

Expanding \( (x - 5)^2 \):

\( (x - 5)^2 = x^2 - 10x + 25 \)

Wait, but maybe the side is \( x - 5 \)? Wait, maybe the diagram has the head's side as \( x - 5 \). So the area of the head is \( (x - 5)^2 = x^2 - 10x + 25 \).

Wait, but maybe I misread the diagram. Wait, the user's diagram: the head is a square with vertical side \( x - 5 \), so yes, side length \( x - 5 \). So area is \( (x - 5)^2 \).

Step1: Identify the shape and formula

The robot’s body is a rectangle. The formula for the area of a rectangle is \( A = l \times w \), where \( l \) is the length and \( w \) is the width. From the diagram, the length of the body is \( 4x + 5 \) and the width is \( x + 2 \).

Step2: Substitute the length and width into the formula

\( A = (4x + 5)(x + 2) \)

Step3: Expand the product

Using the distributive property (FOIL method):

\( (4x + 5)(x + 2) = 4x \times x + 4x \times 2 + 5 \times x + 5 \times 2 \)

\( = 4x^2 + 8x + 5x + 10 \)

\( = 4x^2 + 13x + 10 \)

Step1: Recall the areas of head and body

From Question 7, the area of the head is \( (x - 5)^2 = x^2 - 10x + 25 \).

From Question 8, the area of the body is \( 4x^2 + 13x + 10 \).

Step2: Add the two areas to find the total area

Total area \( = \) Area of head \( + \) Area of body

\( = (x^2 - 10x + 25) + (4x^2 + 13x + 10) \)

Step3: Combine like terms

Combine \( x^2 \) terms: \( x^2 + 4x^2 = 5x^2 \)

Combine \( x \) terms: \( -10x + 13x = 3x \)

Combine constant terms: \( 25 + 10 = 35 \)

So total area \( = 5x^2 + 3x + 35 \)

Wait, but let's check the expansion of the head's area again. Wait, maybe I made a mistake in the head's side length. Wait, the diagram: the head is a square, and the vertical side is \( x - 5 \), but maybe the side is \( x - 5 \)? Wait, but if \( x - 5 \) is the side, then area is \( (x - 5)^2 \). But let's verify with the body. Wait, the body is a rectangle with length \( 4x + 5 \) and width \( x + 2 \), so area is \( (4x + 5)(x + 2) = 4x^2 + 8x + 5x + 10 = 4x^2 + 13x + 10 \), which is correct.

Then total area: \( (x^2 - 10x + 25) + (4x^2 + 13x + 10) = 5x^2 + 3x + 35 \).

Answer:

The area of the robot’s head is \( x^2 - 10x + 25 \) (or \( (x - 5)^2 \)).

Question 8: