A child whose weight is 292 N slides down a 5.20 m playground slide that makes an angle of 35.0° with the horizontal. The coefficient of kinetic friction between slide and child is 0.130. (a) How much energy is transferred to thermal energy? (b) If she starts at the top with a speed of 0.368 m/s, what is her speed at the bottom?

To answer the first question, we need to calculate the work done by friction, which will be equal to the energy transferred to thermal energy. The work done by friction can be calculated using the formula:

Work = Force of friction * distance * cos(theta)

Where:
- Force of friction = coefficient of kinetic friction * normal force
- distance = length of the slide
- theta = angle between the slide and the horizontal plane

First, let's calculate the normal force acting on the child. The normal force is equal to the weight of the child, which can be calculated using the formula:

Weight = mass * gravity

Where:
- mass = weight / gravity
- gravity = 9.8 m/s^2 (acceleration due to gravity)

For the given weight of the child (292 N), the mass will be:
mass = 292 N / 9.8 m/s^2 = 29.8 kg

Now, let's calculate the normal force:
normal force = weight = 292 N

Next, we can calculate the force of friction:
force of friction = coefficient of kinetic friction * normal force
force of friction = 0.130 * 292 N = 37.96 N

Now, let's calculate the work done by friction:
work = force of friction * distance * cos(theta)
work = 37.96 N * 5.20 m * cos(35.0°)

To calculate cos(35.0°), we need to convert the angle to radians:
cos(35.0°) = cos(35.0° * π/180) ≈ 0.819

work ≈ 37.96 N * 5.20 m * 0.819 ≈ 160 J

Therefore, the energy transferred to thermal energy is approximately 160 Joules.

Now, let's move on to the second question.

We can use the principle of conservation of energy to calculate the child's speed at the bottom of the slide. The total mechanical energy at the top of the slide (initial position) is equal to the total mechanical energy at the bottom of the slide (final position). The total mechanical energy is the sum of kinetic energy and potential energy.

At the top of the slide:
- Kinetic energy = 1/2 * mass * velocity^2
- Potential energy = mass * gravity * height
- Total mechanical energy = Kinetic energy + Potential energy

At the bottom of the slide:
- Kinetic energy = 1/2 * mass * velocity^2
- Potential energy = mass * gravity * height
- Total mechanical energy = Kinetic energy + Potential energy

Since the child starts at the top with a speed of 0.368 m/s and the height of the slide is not given, we'll assume the height is negligible compared to the length of the slide.

Therefore, the kinetic energy at the top and bottom of the slide should be the same.

Let's use this information to calculate the speed at the bottom of the slide.

Kinetic energy at the top = Kinetic energy at the bottom
1/2 * mass * (0.368 m/s)^2 = 1/2 * mass * velocity^2

Simplifying the equation:
(0.368 m/s)^2 = velocity^2

Taking the square root of both sides:
0.368 m/s ≈ velocity

Therefore, the speed of the child at the bottom of the slide will be approximately 0.368 m/s.