Thread: I need lots of help, please!

I've never been good in math, but I have to do good on these problems for my class; I've tried solving them myself and am giving myself a headache just trying to figure out what formulas to use. I could really use some help!!

1. Determine whether the following argument is correct. If it's not correct, explain what is wrong with the argument and change the minor premise to make a correct argument.
If a traingle is equilateral, then it has three equal sides.
Triangle ABC doesn't have three equal sides.
Therefore, triangle ABC isn't equilateral.

2. A city block 500 feet by 500 feet is an unobstructed paved lot, except for a small office building 100 feet by 100 feet, centered in the middle of the block. What's the shortest distance from the SW corner to the NE corner, going through the paved lot and along (but not through) the building (to the nearest foot)?

3. Suppose a 12-foot ladder is leaning against a wall so that its base is 2 feet from the edge of the wall. As you climb to the very top of the ladder, you find you can reach 7 feet above the top of the ladder. To what height are you reaching (in feet, to one decimal place)?

4. Find the dimensions of a rectangular region with maximum area that can be enclosed by 80 feet of fencing.

5. The Fibonacci sequence of numbers begins 1,1,2,3,5,8,13,21,34,55,... and continues by the rule: Every number is the sum of the two previous numbers. Determine the next five (eleventh through fifteenth) elements of this sequence. Compute the ratio of the fourteenth and fifteenth elements of this sequence. Is it close to the Golden Ratio?

Please help if you can!! I have to show work too!

Originally Posted by chickenwing

1. Determine whether the following argument is correct. If it's not correct, explain what is wrong with the argument and change the minor premise to make a correct argument.
If a traingle is equilateral, then it has three equal sides.
Triangle ABC doesn't have three equal sides.
Therefore, triangle ABC isn't equilateral.

A statement and its contrapositive are logically equivalent so let's look at the contrapositive version of the statement:
If a triangle does not have three sides, it is not equilateral.

We know triangle ABC does not have three sides, so thus it is not equilateral.

Thus there is nothing wrong with the argument.

-Dan

Originally Posted by chickenwing

Suppose a 12-foot ladder is leaning against a wall so that its base is 2 feet from the edge of the wall. As you climb to the very top of the ladder, you find you can reach 7 feet above the top of the ladder. To what height are you reaching (in feet, to one decimal place)?

You'll have to sketch it, but the ladder and corner of the wall-floor make a right triangle with base (floor) 2 feet and hypotenuse 12 feet. We need to find the length of the wall side.

So. It's a right triangle, so w^2 + 2^2 = 12^2 or w = sqrt(12^2 - 2^2) = sqrt(140).

Thus the height is sqrt(140) + 7, which is approximately 18.8 feet.

-Dan

Originally Posted by chickenwing

Find the dimensions of a rectangular region with maximum area that can be enclosed by 80 feet of fencing.

Call the dimensions of the rectangle x and y. Thus P = 2x + 2y = 80 and A = xy.

Thus y = (1/2)(80 - 2x) = 40 - x.

Thus A = xy = x(40 - x) = -x^2 + 40x.

I don't know if you know Calculus so I'll do it without Calculus. We wish to maximize the area function. Note that this is the form of a parabola that opens downward. To find the maximum we need to find the vertex point. The vertex of the parabola y = ax^2 + bx + c will be on the line x = -b/(2a), so the vertex of the area parabola above will be on the line x = -40/(2*-1) = 20. This will be the x value of the maximum area so x = 20 ft., y = 40 ft. - 20 ft. = 20 ft. are the dimensions of your rectangle. (No it isn't always a square!)

(Doing it by Calculus, A' = -2x + 40. Set this to 0 and solve for x. This gives x = 20 again.)

-Dan

Originally Posted by chickenwing

The Fibonacci sequence of numbers begins 1,1,2,3,5,8,13,21,34,55,... and continues by the rule: Every number is the sum of the two previous numbers. Determine the next five (eleventh through fifteenth) elements of this sequence. Compute the ratio of the fourteenth and fifteenth elements of this sequence. Is it close to the Golden Ratio?

What is "close?" The ratio of the two numbers is 1.618037135 and the Golden Ratio is (1/2)(1 + sqrt(5)) = 1.618033989. The difference between these two numbers is 0.000194433 %. That's close enough for me anyway.

-Dan

Thank you, thank you, thank you!!!!!