Question

1. if the point n(2, 7) is rotated clockwise about the origin, it will produce an image point. match the degrees of rotation with the image of the point. 90° 180° 270° 360° n(7, -2) n(-7, 2) n(-2, -7) n(2, 7) 24. a cone and a cylinder each have the same diameter, and each has a height of 9 inches. the cone is filled with water, and then the water is poured from the cone into the cylinder. what height, in inches, will the water be in the cylinder? write your answer in the box. inches

Explanation:

Question 23

To solve the rotation of point \( N(2, 7) \) about the origin:

• 90° clockwise rotation: The rule is \( (x, y) \to (y, -x) \). So \( (2, 7) \to (7, -2) \), so \( 90^\circ \) matches \( N'(7, -2) \).
• 180° rotation: The rule is \( (x, y) \to (-x, -y) \). So \( (2, 7) \to (-2, -7) \), so \( 180^\circ \) matches \( N'(-2, -7) \).
• 270° clockwise rotation: The rule is \( (x, y) \to (-y, x) \). So \( (2, 7) \to (-7, 2) \), so \( 270^\circ \) matches \( N'(-7, 2) \).
• 360° rotation: A full rotation brings the point back to its original position, so \( 360^\circ \) matches \( N'(2, 7) \).

Step 1: Recall Volume Formulas

The volume of a cone is \( V_{cone} = \frac{1}{3}\pi r^2 h \), and the volume of a cylinder is \( V_{cylinder} = \pi r^2 h \). Given the cone and cylinder have the same diameter (so same radius \( r \)) and height \( h = 9 \) inches.

Step 2: Volume Relationship

The volume of the cone is \( \frac{1}{3} \) the volume of the cylinder with the same radius and height. When we pour the water from the cone to the cylinder, the volume of water (\( V_{water} = V_{cone} \)) will occupy a portion of the cylinder. Let the height of water in the cylinder be \( h_{water} \).

We know \( V_{water} = V_{cone} = \frac{1}{3}\pi r^2 (9) = 3\pi r^2 \).

The volume of the cylinder with water is \( V_{water} = \pi r^2 h_{water} \).

Step 3: Solve for \( h_{water} \)

Set \( \pi r^2 h_{water} = 3\pi r^2 \). Divide both sides by \( \pi r^2 \) (since \( r
eq 0 \)), we get \( h_{water} = 3 \).

Answer:

• \( 90^\circ \): \( N'(7, -2) \)
• \( 180^\circ \): \( N'(-2, -7) \)
• \( 270^\circ \): \( N'(-7, 2) \)
• \( 360^\circ \): \( N'(2, 7) \)

Question 24