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Ryan K.

Certified 6-12 Math Teacher, Tutor for 4 years

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Pre-Calculus

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Question:

There is a centuries-long Indian legend about a local king who was a big chess enthusiast. Legend has it that he would frequently challenge visitors to a game of chess. One time, in order to motivate someone to accept his challenge, the king said that should he lose he would give any reward the visitor wanted. In the end, the king lost, and the visitor asked that the king place rice on a chessboard, putting 1 grain in the first box, 2 in the second, 4 in the third, 8 in the fourth, and so on. Thinking this was a fairly simple reward, the king gave in. He had a bag of rice brought in and he started placing rice on the chessboard. One bag of rice contains 400 grains. Which square will be the first that requires more than one full bag of rice to be filled?

Ryan K.

Answer:

Let's consider writing an equation to represent the amount of rice on each square. Being that the amount of rice is being doubled from square to square, we see the rate of change is not constant, but changing by a factor of two - this tells us that we need an exponential equation of the form $$y = 2^x$$, with $$x$$ representing the number square, and $$y$$ representing the number of grains of rice. Now, the question told us that the first square had one grain of rice on it, but when we substitute 1 for $$x$$, our equation tells us there should be two grains of rice on the square. We can change the exponent of our equation to be the quantity $$(x-1)$$, and that will provide the values we expect. We can now substitute 800 in our equation for $$y$$, and solve for $$x$$. $$800 = 2^{x-1}$$ Logarithmic functions are the inverse of exponential functions, so we can take the $$\log_{2}$$ of both sides of the equation to obtain $$\log_{2}(800) = x - 1$$. Thus $$x = 1 + \log_{2}(800) = 1 + \frac{\log800}{\log2}$$ using the change of base formula. We obtain the result $$x = 9.64$$, which means that our first square that has more than 400 grains will be square 10. We can check our work in two ways, by substituting $$x=9$$ and $$x=10$$ into our original equation and verifying, or by graphing our equation and finding where it intersects $$y=400$$.

Trigonometry

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Question:

A painter is using a 12-foot ladder to paint the side of a house. If the angle the ladder makes with the ground is less than 65°, it will slide out from under him. What is the greatest distance that the bottom of the ladder can be from the side of the house and still be safe for the painter?

Ryan K.

Answer:

Let's imagine this scenario as a right triangle, with the legs being the ground and the wall of the house, and the hypotenuse being the 12-foot ladder. We are told the angle between the ladder and the ground (the hypotenuse and the bottom leg of our triangle) can be no less than 65°. Since 65° is the minimum angle for safety, that means that the distance the bottom of the ladder is from the house at that angle will be the furthest distance the ladder can safely be from the house. We need to find the length of the leg represented by the ground using right angle trigonometry. Since we know the hypotenuse and need to find the side adjacent to the angle we are given, we are going to use the cosine function. SOH-CAH-TOA tells us that cosine is equal to the ratio of adjacent leg length to hypotenuse length, thus: Let A represent the length of the leg of the triangle adjacent to our angle. $$cos(65°) = A/12$$ Multiply both sides of the equation by 12 to isolate the variable A. Using a calculator, we can calculate that $$12*cos(65°) = 5.07$$. That means that the maximum distance the ladder can be from the house is 5.07 feet.

Algebra

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Question:

Is $$(x - 3)$$ a factor of $$2x^4 - 7x^3 - 38x^2 + 103x + 60$$? Explain your reasoning in one to two sentences.

Ryan K.

Answer:

By definition, when a quantity is divided by one of its factors, the remainder of that division will be 0. Thus, if we divide $$2x^4 - 7x^3 - 38x^2 + 103x + 60$$ by $$(x - 3)$$ and get a remainder of 0, then the binomial is in fact a factor. To do this, we can use either polynomial long division or synthetic division. Since we are dividing by a binomial, and it has a leading coefficient of 1, it will be simpler to use synthetic division in this case. When we perform the synthetic division, we obtain a quotient of $$2x^3 - x^2 -41x - 20$$ with a remainder of 0, telling us that $$(x - 3)$$ is in fact a factor.

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