I am volunteering some of my time to help students understand some of the more difficult topics in school. In doing so, I hope that I can help make school much easier.
Question A pond is enclosed by a wooden deck that is 3 feet wide. The fence surrounding the deck is 100 feet long. If the pond is rectangular and the length of the pond is to be three times its width, what are its dimensions?
Solution Let x = width of the pond 3x = length of the pond Perimeter of the rectangle = 100 ft The formula for perimeter is 2x + 6x = 100
Hence, 8x = 100 x = 12.5
Therefore, the width of the pond is 12.50 feet and its length is 37.50 feet.
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Question Mike invested $2000 in bonds and certificate of deposits. the bonds paid 5% and the CD's paid 4.5% if his total interest earned was $93.75 for the year, how much did he invest in each type?
Solution Let x = investment in bonds 2000-x = investment in certificate of deposits
Therefore, .05x + .045(2000-x) = 93.75
Hence, x = $750
Therefore, Mike needs to invest $750 in bonds and $1,250 in certificate of deposits.
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Question A company sells shoes for $40 per pair if less than 50 pair are ordered. If more than 50 are ordered (up to 600), the price is reduced at a rate of $.04 times the number oredered. How many pairs can a dealer purchase for $8400
Solution Let x = number of shoes which can be purchased by $8,400 Therefore, ($40-$0.04x)x = 8,400 40x - 0.04x^2 = 8400
The above equation is a simple quadratic equation.
Divide both sides of the equation by 0.04 to get 1,000x - x^2 = 210,000 Multiply both sides by -1 to get x^2 - 1000x = -210000 Transform the above quadratic equation to get x^2 - 1000x + 210000 = 0 Solve for the roots of the quadratic equation by thinking of two numbers when multiplied equals 210,000 and when added equals 1000 --- 700 and 300!
If you test the above numbers by substituting them in the quadratic equation 40x - 0.04x^2 = 8400, both numbers arrive at $8,400. This means that if you buy 300 pairs of shoes, you will pay $8,400 AND if you buy 700 pairs of shoes, you will still pay $8,400!
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Question Need to know how much of 4200 investment was at 15% and how much of 4200 investment was at 8% to equal profit of 448.
Solution If you have available funds for investment and you want to create an investment plan in order for you to achieve a certain return on investment, then there is a need to allocate your investments.
For the investment problem above, the target return on investment of $4,200 is $448 and the investment portfolio is comprised of an investment with a return of 15% and another with a return of 8%.
So, how do we compute for the funds to be allocated to each investment option?
Let x = investment with a return of 15% (4,200-x) = investment with a return of 8%
Therefore, the expected return on investment is .15x + .08(4,200-x) = 448
Simplify .07x + 336 = 448
Isolate x x = $1,600
Therefore, the investor needs to invest $1,600 in the investment with an expected return of 15% and $2,600 in the investment with an expected return of 8% to achieve a portfolio rate of return of 10.67% or $448.
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Question Mary left by plane to visit her mom in LA, 420km away. Fifteen minuntes later, her mom left to meet her at the airport. She drove 20km to the airport at 40km per hr, arriving just as the plane taxied in. What was the speed of the plane?
Solution First, speed is calculated by D ---- t
where D = distance t = time
Second, determine the distance traveled by the airplane. 420 km
Third, determine the time spent by the airplane to travel such distance 1. 15 minutes after take off, Mary's mother set out for the airport 2. It took 30 minutes for Mary's mother to travel the 20 kilometer distance to the airport at a speed of 40 kilometers per hour.
Total travel time for the airplane therefore is 45 minutes.
Lastly, calculate for the speed 420 kilometers ----------------------- 45 minutes
The speed of the airplane is 9.33 kilometers per minute OR 560 kilometers per hour.
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Question If the perimeter of a rectangular block of land is 46 meters, find the lenght and width of the block.
Solution The formula for a rectangle's perimeter is 2L + 2W where L is length and W is width. Hence the above problem can be written as 2L + 2W = 46 which is also equal to L + W = 23, however it can not be solved yet given the information. The relationship of the rectangle's width and length must be stated in rder for the problem to be solved.
Question If the length of a rectangular parking lot is 10 meters less than twice the width, and the perimeter is 100 meters, find the length of the parking lot
Solution First, write the word problem as an equation. Let x = width of the rectangle Then 2x - 10 = length of the rectangle Then its perimeter is 2x + 2(2x - 10) = 100
The perimeter of a rectangle is computed as 2L + 2W
Simplify the above formula 2x + 4x - 20 = 100 ===> 6x = 120 Therefore x = 20
The length of the rectangle is 30 AND its width is 20.
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Question The perimeter of the quadrilateral is 120cm. Find the lengths and sides. (x+5)(x^2-3)(3x-8)(x+3).
Solution Let us first define how to measure the perimeter of a quadrilateral. The perimeter of any quadrilateral, or any shape with four sides for that matter, is measured by adding the lengths of its sides. Hence, the quadrilateral on the right has a perimeter of 30 which is computed by adding the measure of its sides - 6 + 9 + 7 + 8.
Now, let us go back to our problem. The four sides of the quadrilateral are (x+5), (x^2-3), (3x-8) and (x+3). Hence, its perimeter is computed by (x+5) + (x^2-3) + (3x-8) + (x+3).
To solve for the length of the sides of the quadrilateral:
(x+5) + (x^2-3) + (3x-8) + (x+3) = 120 x^2 - 3+ x+5 + 3x - 8 + x + 3 = 120 x^2 + 5x - 3 = 120 ===> add like terms x^2 + 5x - 123 = 0 ===> right side of the equation is set to zero (x + ?) (x - ?) = 0 ====> to arrive at the root of the quadratic equation, think of two numbers that when you multiply them yields -123 and when you add them gives 5. Hmmm...difficult, however nothing prevents us to looking at a solution which is NOT a whole number.
Now, I got 13.87 and -8.87. Their multiple is -123.02 [close enough] and their sum is 5.
(x + 13.87) (x - 8.87) = 0 x = -13.87 ===> this is NOT an option because it is NEGATIVE. x = 8.87
Now to calculate the lengths of the sides of the quadrilateral: (x+5) ===> 13.87 (x^2-3) ==> 75.68 (3x-8) ===> 18.61 (x+3) ===> 11.87
Now to check whether the sum of the above is 120. 13.87 + 75.68 + 18.61 + 11.87 = 120.02 [close enough]
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Question When Paula played softball, she struck out twice, walked 3 times and had 6 hits. The probability she will get a hit the next time she is at bat is ____ out of ______.
Solution First, count the number of times Paula played or she was at bat. ANSWER: 11 times Second, count the number of times she had a hit. ANSWER: 6 times Hence, the probability that Paula will have a hit next time she is at bat is 6/11 or 0.55 OR 55 out of 100 OR 55 per cent probability.
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Question If a certain fraction is multiplied by 4 and the result subtracted from twice the reciprocal of the original fraction there remains 1/2 of the original fraction. find the original fraction.
Solution Let x = missing fractions Then, the above word problem can be states as 2*(1/x) - 4x = (1/2)*x Solve for x by isolating all variables with x. Hence, 2/x - 4x - x/2 = 0. Simplify 2/x - 4x - x/2. (2/x - 4x - x/2)*2x = 0 (4 - 8x^2 - x^2)/2x = 0 (4 - 9x^2) = 0 (-3x + 2) * (3x +2) = 0 Then, look for x (-3x + 2)= 0 =====> -3x = -2 =====> x = 2/3 (3x +2) = 0 =====> 3x = -2 =====> x = -2/3
Now, test the validity of the two solutions above by substituting them in the formula 2*(1/x) - 4x = (1/2)*x.
Question Nicole rides her bike every day. If she rides at a constant speed of 10 mph every time she rides, how many miles will she ride if she rides for 30 minutes on Monday, 45 minutes on Tuesday, and 20 minutes on Wednesday?
Solution Monday 30minutes x 10mph = 5.0 miles Tuesday 45minutes x 10mph = 7.5 miles Wednesday 20minutes x 10mph = 3.3 miles
Total miles ridden: 15.8 miles
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Question John's old '87 Lebaron has a 15 gal gas tank and gets 23 mpg. The number of miles he can drive is a function of how much gas is in the tank. Require (a) write this relationship in equation form and (b) determine the domain and range of the function in the context.
Solution Let x = gallons of gas in the tank y = number of miles
Therefore, y = 23x which is bounded by the following ranges:
0 <= x <= 23 for the amount of gas in the tank can not go below 0 and above 15 gallons
0 <= y <= 345 for the number of miles John can travel can not go below 0 and above 345 miles (23 mpg multiplied by 14 gallons)
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Question Jane Manning, an outside business woman, uses her car forboth business and pleasure. Last year, she traveled 30,000 miles using 900 gallons of gas. Her car gets 40 miles per gallon on the Highway, and 25 mpg in the city. She candeduct all highway travel, but no city travel, on her taxes.
How many miles should Jane be allowed as a business expense?
Solution
Letx=gas spent in the Highway
y=gas spent in the City
Therefore, Equation 1: x + y = 900
Therefore,
Equation 2: 40x + 25y = 36,000
Transform
Equation 1to x = 900 – y
Substitute the above equation in Equation 2
Which is equal to 36,000 – 40y + 25y = 30,000
Simplify the above equation to 6,000 = 15y
Hence, y = 400 AND x = 500
Therefore, Jane can deduct 500 gallons OR 10,000 miles on her taxes.
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