Question

directions: find the value of x based on the information given for each shape.
perimeter = 12
triangle with hypotenuse 5.0, leg 3, leg x
12.

1. find the volume and surface area of the sh

3d shape with dimensions 6 cm, 5 cm, 7 cm
section 4
directions: use the protractor to determine each angle.
protractor diagrams

1. ∠ =
2. ∠ =

Explanation:

Problem 12: Find the value of \( x \) in the right triangle

Step 1: Identify the triangle type

This is a right triangle, so we use the Pythagorean theorem: \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse, and \( a, b \) are the legs. Here, \( c = 5.0 \), \( b = 3 \), and \( a = x \).

Step 2: Apply the Pythagorean theorem

Substitute the known values: \( x^2 + 3^2 = 5.0^2 \)
Simplify: \( x^2 + 9 = 25 \)

Step 3: Solve for \( x^2 \)

Subtract 9 from both sides: \( x^2 = 25 - 9 = 16 \)

Step 4: Solve for \( x \)

Take the square root: \( x = \sqrt{16} = 4 \)

Problem 14: Volume and Surface Area of the Prism (Assumed to be a rectangular prism or a triangular prism? Wait, the diagram shows a prism with a rectangular base? Wait, the dimensions are 6 cm, 5 cm, 7 cm, and another 5 cm, 6 cm. Wait, maybe it’s a triangular prism? Wait, no, the base might be a rectangle? Wait, no, the diagram has a right prism with a rectangular base? Wait, no, the sides: 6 cm, 5 cm, 7 cm, and the other face is 5 cm, 6 cm. Wait, maybe it’s a rectangular prism? Wait, no, the prism has a base that is a rectangle? Wait, no, the given dimensions: length \( l = 6 \) cm, width \( w = 5 \) cm, height \( h = 7 \) cm? Wait, no, maybe it’s a triangular prism? Wait, the diagram shows a prism with a triangular base? Wait, no, the user’s diagram: “6 cm, 5 cm, 7 cm, 5 cm, 6 cm” – maybe it’s a rectangular prism (cuboid) with length 6, width 5, height 7? Wait, no, the presence of right angles suggests a rectangular prism (cuboid) or a triangular prism. Wait, let’s assume it’s a rectangular prism (cuboid) with length \( l = 6 \) cm, width \( w = 5 \) cm, height \( h = 7 \) cm.

Volume of a rectangular prism: \( V = l \times w \times h \)

Step 1: Substitute values

\( V = 6 \times 5 \times 7 \)

Step 2: Calculate

\( V = 210 \, \text{cm}^3 \)

Surface Area of a rectangular prism: \( SA = 2(lw + lh + wh) \)

Step 1: Substitute values

\( SA = 2[(6 \times 5) + (6 \times 7) + (5 \times 7)] \)

Step 2: Calculate each term

\( 6 \times 5 = 30 \), \( 6 \times 7 = 42 \), \( 5 \times 7 = 35 \)

Step 3: Sum the terms

\( 30 + 42 + 35 = 107 \)

Step 4: Multiply by 2

\( SA = 2 \times 107 = 214 \, \text{cm}^2 \)

Problem 15: Measure the angle with a protractor

The protractor shows the angle between the horizontal line and the other ray. The arrow points to 150°? Wait, no, the protractor scale: the outer scale (if the horizontal line is 0° on the right) – wait, the horizontal line is at 0° (right), and the other ray is at 150°? Wait, no, looking at the protractor: the horizontal line is the base, and the other ray is at 150°? Wait, no, the first protractor (problem 15) has the horizontal line (right) as 0°, and the other ray is at 150°? Wait, no, the markings: the protractor has 0° on the right, and the ray is at 150°? Wait, no, the angle is measured from the horizontal (right) to the ray. Let’s check the scale: the protractor has numbers from 0 to 180. The ray is at 150°? Wait, no, the arrow is at 150? Wait, the first protractor (problem 15) – the angle is 150°? Wait, no, maybe 150°? Wait, the horizontal line is 0° (right), and the ray is at 150°? Wait, no, the angle is 150 degrees? Wait, let’s see: the protractor’s inner scale or outer scale. If the horizontal line is 0° (right), and the ray is at 150° (outer scale), then the angle is 150°.

Problem 16: Measure the angle with a protractor

The second protractor (problem 16) has the horizontal line (right) as 0°, and the ray is at a small angle, maybe 10° or 15°? Wait, the protractor scale: the outer scale (if the horizontal line is 0° on the right) – the ray is near 10°? Wait, the markings: the protractor has 0° on the right, and the ray is at 10°? Wait, the angle is 10°?

Answer:

s:

1. \( x = \boldsymbol{4} \)
2. Volume = \( \boldsymbol{210 \, \text{cm}^3} \), Surface Area = \( \boldsymbol{214 \, \text{cm}^2} \) (assuming rectangular prism)
3. \( \angle = \boldsymbol{150^\circ} \) (approximate, based on protractor)
4. \( \angle = \boldsymbol{10^\circ} \) (approximate, based on protractor)