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Kelly P.

HS Math Teacher for 7 years

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Pre-Calculus

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Question:

Solve for all values of x x^3 - 5x^2 + 9x -5 = 0

Kelly P.

Answer:

We start by using the rational roots theorem to identify possible solutions. Since our leading coefficient is 1 and our constant -5, we are looking for factors of 5. So the possibilities are +/- 1, +/-5. To start lets take a guess to see if one of these is a solution. Let's try x=1 We can use synthetic division or just plus in x=1 to see if it is a solution so if we plug in 1 1^3 - 5*1 + 9*1 - 5 = 1 -5 + 9 -5 = 0 So 1 is a solution, which means that (x - 1) must be a factor. We will now use polynomial long division or synthetic division so divide (x - 1) out. 1 1 -5 9 -5 1 -4 5 ----------------- 1 -4 5 0 So we are left with x^2 - 4x + 5 This is now a quadratic and we can solve it with factoring or the quadratic formula. Let us use the quadratic formula 4 +/- sqrt((-4)^2 - (4)(1)(5) all divided by 2 4 +/- sqrt(16 - 20) all divided by 2 4 +/- sqrt(-4) all divided by 2 4 +/- 2i all divided by 2 2 +/- i so our three solutions are 1, 2+ i, 2 - i

Geometry

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Question:

Bob is 6 ft tall and measures his shadow to be 9ft long. He notices that a lamp-post casts a 12ft long shadow. How tall is the lamp post?

Kelly P.

Answer:

We can answer this problem using similar triangles. Since it is the same time of day the shadow cast will be proportion to the height of the object. So we can create proportions Bobs height/Bobs shadow = Lamp height/Lamp shadow 6/9 = x/12 We can solve this by cross multiplying so we get 6x = 108 divide by 6 and we get x = 18. So the building is 18 ft tall.

Statistics

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Question:

Alice asks a random sample of 100 students from her school whether they have been to Canada. She receives 65 answers of yes. Construct a 95% confidence interval for the percent of students in the school who have been to Canada.

Kelly P.

Answer:

First we must check the conditions. 1. # successes (np) > 10 -- This is true since 65>10 2. # failures (nq) > 10 -- This is true since 35>10 3. The problem states that it was a random sample So our conditions are met and we will proceed with a 1-proportion z-test The formula for this is p-hat +/- z*(sqrt(pq/n)). For a 95% confidence interval, we know that z* is 1.96 (using the z-chart or a calculator) so we get .65 +/- 1.96*sqrt(.65*.35/100) = (.548, .743) So then we write our conclusion. I am 95% confident that the true proportion of all students in alice's school who have been to Canada is between .548 and .743

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