# Tutor profile: Joe B.

## Questions

### Subject: Trigonometry

Find the height of an upright telephone pole if you know that the length of the shadow of the telephone pole on level groun is 30 feet, and the angle between the ground to the imaginary line drawn from the tip of the pole's shadow to the top of the pole is 70 degrees.

I love solving these kinds of problems because they have a real world application of the subject material and they involve a bit of interpretation of the information given in the problem statement. It makes the solver think beyond just where they need to plug the given numbers into an equation. In this problem we want to find the height of an upright telephone pole. From the statement "upright pole" we can assume that the pole makes a 90 degree angle with the level ground. Additionally, we know that an object's shadow extends from the base of an object and along the ground. So at this point we know that we have a pole of unknown height, and a shadow that is 30 feet in length that makes a 90 degree angle with our vertical pole. From this information alone we can start to see a right triangle forming. Finally, we also know that an imaginary line drawn between the tip of the pole's shadow and the top of the pole makes a 70 degree angle with the ground. Now we have all three sides of an enclosed right triangle: The vertical pole The horizontal shadow of the pole on the ground The imaginary line from the tip of the shadow to the top of the pole We also have two angles and a side length of the right triangle: 90 degrees between the vertical pole and horizontal shadow 70 degrees between the horizontal shadow and the line between the tip of the shadow and the top of the pole 30 feet between the base of the pole and the tip of the shadow When we sketch this right triangle we see that we have our 70 degree angle and the adjacent side length (30 feet), but we want to find the opposite side length (pole height). Now we can use our trigonometry functions to solve for the height of the pole: tan(theta) = (opposite side length) / (adjacent side length) Where: theta = 70 degrees opposite side length = pole height adjacent side length = 30 feet So, tan(70 degrees) = (pole height) / (30 feet) 30*tan(70 degrees) = pole height pole height = 82.42 feet

### Subject: Calculus

Given the function f(x) = x^3 - 5*x^2 + x - 7 , solve for the slope of the plot of f(x) at x = 5.

The definition of the derivative is the rate of change of a function with respect to a variable. In this case our variable is x = 5. In order to find the slope of the plot at x = 5 we can find the derivative of the given function and solve for x = 5. First we find the derivative of the given function: f'(x) is the derivative of the function f(x) We can find the derivative of each summed term in the function f(x) to find f'(x). Lets start with x^3. In this case the term is equivalent to 1*x^3 so the exponent is 3 and the leading multiplier is 1. To take the derivative of x^3 we will multiply the exponent by the leading multiplier to find the leading multiplier of the term's derivative (3 * 1 = 3). Then we will subtract 1 from the exponent to find the exponent of the term's derivative (3 - 1 = 2). Therefore, the derivative of the term x^3 is 3*x^2. We repeat this process for each summed term in f(x): -5*x^2 => -10*x x => 1*x^0 = 1 -7 => 0 We sum these term to get f'(x) = 3*x^2 - 10*x + 1 + 0 Now that we have the derivative of the given function, we plug in our variable (x = 5) to find the slope at our desired location: f'(5) = 3*(5)^2 - 10*(5) + 1 f'(5) = 75 - 50 + 1 f'(5) = 26 Our solution for the slope of our function f(x) at x = 5 is 26

### Subject: Physics

Describe Newton's laws of motion in your own words and provide a real world situation where each law would apply.

Newton's first law of motion states that a stationary object will remain stationary until acted on by a force, and an object in motion will continue in the same direction and rate of motion until acted on by a force. For example, if a asteroid is traveling through deep space it will maintain the same direction and rate of travel until it is influenced by a force such as a gravitational force or an impact from anther object. Newton's second law of motion states that when a force acts on an object, the change in velocity of the object, or acceleration, follows the relationship F=m*a which can be reorganized as a=F/m. where: F = Magnitude of force acting on the object (Newtons) m = mass of the object (kilograms) a = acceleration of the object (meters per second squared) For example, if we know that our asteroid has a mass of 1,000 kilograms, and experiences a force equivalent to 1,000 Newtons in one direction, then the acceleration of our asteroid will be a = F/m = (1,000N) / (1,000kg) = 1 m/s^2 in the same direction that the force is applied for as long as the force is applied. Newton's third law of motion states that "for every action there is an equal and opposite reaction". This can be interpreted as: for every applied force there is an equal and opposite reaction force. For example, lets assume that the force on the asteroid is performed by an astronaut with a mass of 100kg who is using their arms to push the asteroid with a force of 1,000N. The asteroid is still going to accelerate at 1 m/s^2 according to Newton's second law. However, our astronaut will experience the same force that they are exerting on the asteroid in the opposite direction that the asteroid experiences the force. Therefore, we can use Newton's second law applied to the astronaut to find that the astronaut's acceleration is a = F/m = (1,000N) / (100kg) = 10 m/s^2. Also remember that the astronaut's acceleration will be in the opposite direction of the asteroid's acceleration according to Newton's third law. The accelerations of the asteroid and astronaut will continue for as long as the astronaut is pushing on the asteroid. Once the asteroid and astronaut are separated due to their travel in opposite directions they will each travel in their own directions at a constant rate of motion according to Newton's first law.

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