Geometry

We are given the point . To find the foot of the perpendicular from this point to the y-axis, we observe that the x-coordinate and z-coordinate of the foot of the perpendicular will be 0, as the perpendicular will lie on the y-axis. Hence, the coordinates of the foot of the perpendicular from on the y-axis are . Thus, the correct answer is .

The length of the median is the distance from A to the midpoint of BC. First, we find the midpoint of BC, which is the average of the coordinates of points B and C. Let the coordinates of B and C be and respectively. The midpoint is: Now, the distance from A to M is: Thus, the length of the median is .

We are given that the volume of water leaking is . The negative sign indicates that the volume is decreasing. The container is cylindrical, so the volume of water in the container at any time is given by the formula: where is the radius of the base and is the height of the water. Step 1: Differentiate the volume formula with respect to time : We know that the radius cm, and the height of the water is given as cm (since the height of water is 2 meters, which equals 200 cm). Step 2: Substitute the given values into the equation: Step 3: Solve for : Step 4: Approximate the rate: Thus, the rate at which the height of water is decreasing is approximately .

For a spherical balloon: - Surface area - Volume Given that the surface area increases at a rate of 5 mm²/s, we have: Differentiate the surface area with respect to time: Substitute and mm into this equation: Solving for : Next, we want to find the rate at which the volume is increasing, . Differentiate the volume formula: Substitute mm and : Simplifying: Thus, the rate at which the volume of the balloon is increasing is .

The volume of a hollow sphere is given by the formula: where is the outer radius and is the inner radius. We are given that the volume is constant, which implies: Now, we differentiate both sides of the volume equation with respect to time : Using the chain rule: Simplify: Now, substitute the known values: - (rate of change of the inner radius),
- , and
- .
Substitute these into the equation: Thus, the rate of increase of the outer radius is .