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 test/384085: Interest Formulas If you invest \$P, and you earn interest only on the amount you invested (the \$P), then ________________ interest is earned. If you invest \$P and you earn interest not only on the \$P, but also on the interest gained, then ______________ interest is earned. Let P = principal, r = annual nominal interest rate (in decimal form), t = time of the investment (in years), A = amount in the account after t years. THEN… Simple interest: A = _________________________________________ Compound interest: A = _____________________________________ Ideally, what do you want to happen to the value of n in the compound interest formula if the formula represented the amount of money you have in the bank?1 solutions Answer 271873 by Theo(3464)   on 2010-12-10 08:05:20 (Show Source): You can put this solution on YOUR website!If you invest \$P, and you earn interest only on the amount you invested (the \$P), then SIMPLE interest is earned. If you invest \$P and you earn interest not only on the \$P, but also on the interest gained, then COMPOUND interest is earned. Let P = principal, r = annual nominal interest rate (in decimal form), t = time of the investment (in years), A = amount in the account after t years. THEN… SIMPLE INTEREST A = P + (P * r * t) Example: P = 10,000 r = .10 t = 5 years A = 10,000 + (10,000 * .10 * 5) = 10,000 + 5,000 = 15,000 COMPOUND INTEREST A = P * (1+r)^t Example: P = 10,000 r = .10 t = 5 years A = 10,000 * (1.10)^5 = 10,000 * 1.61051 = 16,1051 You make more money with compound interest than you do with simple interest, assuming the same investment in the same time frame with the same rate of return. LAST QUESTION Ideally, what do you want to happen to the value of n in the compound interest formula if the formula represented the amount of money you have in the bank? I don't see "n" anywhere in your question. If "n" is the number of compounding periods per year, than, ideally, you would want n to be as large as possible. If n = 1, then the number of compounding periods per year is 1. If n = 12, then the number of compounding periods per year is 12. When n is greater than 1, you multiply the number of years by n and you divide the nominal interest rate by n. The nominal interest rate is the annual interest assuming only 1 compounding period per year. In the compounding example above, n was equal to 1 (not shown in the formula. with n in the formula, the formula becomes: A = P * (1 + (r/n))^(t*n) When n was equal to 1, you got: A = P * (1 + r) ^ n That's what we solved earlier. When n is 12, the formula becomes: A = P * (1 + (r/12))^(t*12) Same compounding example but with number of compounding periods per year equal to 12. P = 10,000 r = .1 t = 5 r/12 = .1/12 = .008333333333 t*12 = 5*12 = 60 formula becomes: A = 10,000 * (1.0083333333) ^ 60 = 16453.08934 with monthly compounding, the future value is 16,453.08934 with yearly compounding, the future value is 16,16,105.1 Your savings are greater when the number of compounding periods per year are greater. The most compounding periods per year you can get is with continuous compounding. That formula is A = P * e^rt) e is the well known scientific constant of 2.718281828..... Here you use the annual interest rate and the number of years. You get: A = 10,000 * e^(.1*5) = 10,000 * e^.5 = 16,487.21271 That's the theoretical maximum number of compounding periods per year. If you compounded daily, you would get close to that. Assume daily compounding with 365 days per year. P = 10,000 i = .1 / 365 = .000273973 t = 5 * 365 = 1825 A = P * (1.000273973)^1825 = 16486.08362 That's pretty close to the theoretical maximum, but not exactly. Bottom Line is the more the compounding periods per year, the greater your savings will be. Without knowing what you meant by "n", this is the best I can come up with based on what I know about compounding. I can't think of what else "n" can be, given the formulas as presented.
 Polynomials-and-rational-expressions/384076: I don't understand this part. Where do you use the common factor in an expression consisting of various terms? The actual question is What is a common factor? Where do you use the common factor in an expression consisting of various terms? What is the Greatest Common Factor? 1 solutions Answer 271867 by Theo(3464)   on 2010-12-10 07:39:09 (Show Source): You can put this solution on YOUR website!A common factor is a factor that can be divided evenly into each of the terms of an expression. Take the expression 9 + 27 + 54. 3 can be divided evenly into 9 and into 27 and into 54, so 3 is a common factor. 9 can be divided evenly into 9 and into 27 and into 54, so 9 is also a common factor. The greatest common factor is 9 because that is the largest factor that can be divided evenly into each of the terms of the expression. You use the common factor in an expression consisting of various terms in order to simplify the expression so that it can be solved easier. Consider the expression (3x^3 - 17x^2 + 18x - 12) / (x^2 - 5x + 6) Consider that you know that the numerator in this expression is equal to (x^2 - 5x + 6) * (3x-2) The expression becomes ((3x-2) * (x^2 - 5x + 6)) / (x^2 - 5x + 6). the common factor of (x^2 - 5x + 6) cancels out and you are left with a result of (3x-2). Knowing what the common factor is made the division much simpler than not knowing what the common factor is.
 Miscellaneous_Word_Problems/384063: I greatly NEED some help URGENTLY with this problem. i tried to do it forever but i dont get it so would somebody PLEASE HELP ME! I REALLY NEED A QUICK ANSWER. THANK YOU! Lesson 4—4 Two loan balances can be approximated by the equations given in the table. The time x is in years. i will separate things in table with a (/) Loan Amount / Interest rate / Years / Equation \$200,000 / 7% / 20 yrs / y= -9,734.7x + 218,761 \$250,000 / 6.25% / 15 yrs / y= -16,474x + 266,478 1)Use Matrices and the inverse matrix to find when the loan balances will be approximately the same for the two loans before the final payment is made. 2)What will the approximate balance of each loan be at that time? 3)Why might your estimate of the time when the loan balances will be the same be somewhat inaccurate? THANK YOU SO MUCH IN ADVANCE! 1 solutions Answer 271855 by Theo(3464)   on 2010-12-10 04:48:56 (Show Source): You can put this solution on YOUR website!The 2 equations that approximate the solution to this are: y = -9734.7 * x + 218,761 y = -16474 * x + 266,478 If you graph both these equations, you will see when they cross. The graph of both of these equations is shown below: You can see from the graph that the approximate intersection point is somewhere around 7 years. You can calculate that by making the equations equal to each other. When you do that, you are saying that the value of y in the first equation is equal to the value of y in the second equation. you get: -9734.7x + 218761 = -16474 * x + 266,478 Solve this equation algebraically to get an answer of: x = 7.080408945 y = 149835.343 The approximation given by the graph is pretty close. x is the number of years when the balances will be approximately equal. y i the outstanding balance in each account at that time. The estimate will be somewhat inaccurate because the declining balance equation is not a straight line. What the equations show are probability the best fit of a straight line to the actual balances as they drop in time. The actual point where the balances are equal can be calculated through the use of Present Value of A Loan Amount Formulas, or through the use of a mechanized program that will do the calculations for each year of each loan. I used Excel to do that for you, after having calculated what the monthly payments would be using a financial calculator, and the results are shown below: ``` Remaining Balance year Loan1 Loan2 0 \$200,000.00 \$250,000.00 1 \$199,616.07 \$249,158.53 2 \$199,229.90 \$248,312.67 3 \$198,841.47 \$247,462.41 4 \$198,450.79 \$246,607.72 5 \$198,057.82 \$245,748.58 6 \$197,662.56 \$244,884.96 7 \$197,264.99 \$244,016.84 8 \$196,865.10 \$243,144.21 9 \$196,462.89 \$242,267.03 10 \$196,058.32 \$241,385.28 11 \$195,651.40 \$240,498.93 12 \$195,242.10 \$239,607.98 13 \$194,830.41 \$238,712.38 14 \$194,416.33 \$237,812.11 15 \$193,999.82 \$236,907.16 16 \$193,580.89 \$235,997.50 17 \$193,159.52 \$235,083.09 18 \$192,735.68 \$234,163.93 19 \$192,309.38 \$233,239.97 20 \$191,880.58 \$232,311.21 21 \$191,449.29 \$231,377.60 22 \$191,015.48 \$230,439.14 23 \$190,579.14 \$229,495.79 24 \$190,140.25 \$228,547.52 25 \$189,698.80 \$227,594.31 26 \$189,254.78 \$226,636.14 27 \$188,808.17 \$225,672.98 28 \$188,358.95 \$224,704.81 29 \$187,907.12 \$223,731.59 30 \$187,452.64 \$222,753.30 31 \$186,995.52 \$221,769.91 32 \$186,535.73 \$220,781.41 33 \$186,073.26 \$219,787.75 34 \$185,608.09 \$218,788.92 35 \$185,140.20 \$217,784.89 36 \$184,669.59 \$216,775.63 37 \$184,196.23 \$215,761.11 38 \$183,720.11 \$214,741.31 39 \$183,241.21 \$213,716.20 40 \$182,759.52 \$212,685.75 41 \$182,275.02 \$211,649.93 42 \$181,787.69 \$210,608.72 43 \$181,297.52 \$209,562.08 44 \$180,804.50 \$208,509.99 45 \$180,308.59 \$207,452.42 46 \$179,809.79 \$206,389.35 47 \$179,308.09 \$205,320.74 48 \$178,803.45 \$204,246.56 49 \$178,295.87 \$203,166.78 50 \$177,785.34 \$202,081.39 51 \$177,271.82 \$200,990.34 52 \$176,755.31 \$199,893.60 53 \$176,235.78 \$198,791.16 54 \$175,713.23 \$197,682.97 55 \$175,187.62 \$196,569.02 56 \$174,658.95 \$195,449.25 57 \$174,127.20 \$194,323.66 58 \$173,592.34 \$193,192.21 59 \$173,054.37 \$192,054.86 60 \$172,513.25 \$190,911.59 61 \$171,968.98 \$189,762.36 62 \$171,421.53 \$188,607.15 63 \$170,870.90 \$187,445.92 64 \$170,317.05 \$186,278.65 65 \$169,759.96 \$185,105.29 66 \$169,199.63 \$183,925.82 67 \$168,636.03 \$182,740.21 68 \$168,069.14 \$181,548.43 69 \$167,498.95 \$180,350.44 70 \$166,925.43 \$179,146.20 71 \$166,348.56 \$177,935.70 72 \$165,768.33 \$176,718.89 73 \$165,184.72 \$175,495.74 74 \$164,597.70 \$174,266.23 75 \$164,007.25 \$173,030.31 76 \$163,413.36 \$171,787.95 77 \$162,816.01 \$170,539.12 78 \$162,215.17 \$169,283.79 79 \$161,610.83 \$168,021.92 80 \$161,002.96 \$166,753.48 81 \$160,391.55 \$165,478.43 82 \$159,776.57 \$164,196.74 83 \$159,158.00 \$162,908.37 84 \$158,535.82 \$161,613.29 85 \$157,910.02 \$160,311.47 86 \$157,280.56 \$159,002.87 87 \$156,647.43 \$157,687.45 88 \$156,010.61 \$156,365.18 * 89 \$155,370.07 \$155,036.03 * 90 \$154,725.80 \$153,699.95 91 \$154,077.77 \$152,356.92 92 \$153,425.96 \$151,006.88 93 \$152,770.35 \$149,649.82 94 \$152,110.91 \$148,285.69 95 \$151,447.63 \$146,914.45 96 \$150,780.47 \$145,536.08 97 \$150,109.43 \$144,150.52 98 \$149,434.47 \$142,757.75 99 \$148,755.57 \$141,357.72 100 \$148,072.71 \$139,950.40 101 \$147,385.87 \$138,535.75 102 \$146,695.03 \$137,113.73 103 \$146,000.15 \$135,684.31 104 \$145,301.22 \$134,247.44 105 \$144,598.21 \$132,803.09 106 \$143,891.10 \$131,351.22 107 \$143,179.87 \$129,891.78 108 \$142,464.49 \$128,424.74 109 \$141,744.93 \$126,950.06 110 \$141,021.18 \$125,467.71 111 \$140,293.21 \$123,977.63 112 \$139,560.99 \$122,479.79 113 \$138,824.49 \$120,974.14 114 \$138,083.71 \$119,460.66 115 \$137,338.60 \$117,939.29 116 \$136,589.14 \$116,410.00 117 \$135,835.31 \$114,872.75 118 \$135,077.09 \$113,327.49 119 \$134,314.44 \$111,774.18 120 \$133,547.34 \$110,212.78 121 \$132,775.77 \$108,643.25 122 \$131,999.70 \$107,065.54 123 \$131,219.10 \$105,479.61 124 \$130,433.94 \$103,885.43 125 \$129,644.21 \$102,282.94 126 \$128,849.87 \$100,672.11 127 \$128,050.90 \$99,052.89 128 \$127,247.26 \$97,425.23 129 \$126,438.94 \$95,789.10 130 \$125,625.90 \$94,144.44 131 \$124,808.12 \$92,491.22 132 \$123,985.57 \$90,829.39 133 \$123,158.22 \$89,158.90 134 \$122,326.05 \$87,479.71 135 \$121,489.02 \$85,791.78 136 \$120,647.11 \$84,095.05 137 \$119,800.29 \$82,389.49 138 \$118,948.52 \$80,675.05 139 \$118,091.79 \$78,951.67 140 \$117,230.06 \$77,219.32 141 \$116,363.31 \$75,477.95 142 \$115,491.49 \$73,727.50 143 \$114,614.60 \$71,967.94 144 \$113,732.58 \$70,199.22 145 \$112,845.43 \$68,421.28 146 \$111,953.09 \$66,634.09 147 \$111,055.56 \$64,837.58 148 \$110,152.78 \$63,031.72 149 \$109,244.74 \$61,216.45 150 \$108,331.40 \$59,391.73 151 \$107,412.74 \$57,557.51 152 \$106,488.72 \$55,713.73 153 \$105,559.30 \$53,860.35 154 \$104,624.47 \$51,997.31 155 \$103,684.18 \$50,124.58 156 \$102,738.41 \$48,242.08 157 \$101,787.11 \$46,349.79 158 \$100,830.27 \$44,447.64 159 \$99,867.85 \$42,535.58 160 \$98,899.82 \$40,613.56 161 \$97,926.14 \$38,681.53 162 \$96,946.77 \$36,739.44 163 \$95,961.70 \$34,787.23 164 \$94,970.88 \$32,824.86 165 \$93,974.28 \$30,852.27 166 \$92,971.86 \$28,869.40 167 \$91,963.60 \$26,876.20 168 \$90,949.46 \$24,872.63 169 \$89,929.40 \$22,858.61 170 \$88,903.39 \$20,834.11 171 \$87,871.39 \$18,799.06 172 \$86,833.38 \$16,753.42 173 \$85,789.31 \$14,697.12 174 \$84,739.15 \$12,630.11 175 \$83,682.86 \$10,552.33 176 \$82,620.41 \$8,463.74 177 \$81,551.77 \$6,364.26 178 \$80,476.89 \$4,253.85 179 \$79,395.74 \$2,132.45 180 \$78,308.28 (\$0.00) 181 \$77,214.48 182 \$76,114.30 183 \$75,007.71 184 \$73,894.65 185 \$72,775.11 186 \$71,649.03 187 \$70,516.39 188 \$69,377.13 189 \$68,231.24 190 \$67,078.65 191 \$65,919.35 192 \$64,753.28 193 \$63,580.41 194 \$62,400.70 195 \$61,214.10 196 \$60,020.59 197 \$58,820.11 198 \$57,612.63 199 \$56,398.10 200 \$55,176.50 201 \$53,947.76 202 \$52,711.86 203 \$51,468.75 204 \$50,218.38 205 \$48,960.73 206 \$47,695.73 207 \$46,423.36 208 \$45,143.56 209 \$43,856.30 210 \$42,561.53 211 \$41,259.21 212 \$39,949.29 213 \$38,631.73 214 \$37,306.49 215 \$35,973.51 216 \$34,632.76 217 \$33,284.18 218 \$31,927.74 219 \$30,563.39 220 \$29,191.08 221 \$27,810.76 222 \$26,422.39 223 \$25,025.93 224 \$23,621.31 225 \$22,208.51 226 \$20,787.46 227 \$19,358.12 228 \$17,920.45 229 \$16,474.38 230 \$15,019.89 231 \$13,556.90 232 \$12,085.39 233 \$10,605.29 234 \$9,116.56 235 \$7,619.14 236 \$6,112.98 237 \$4,598.05 238 \$3,074.27 239 \$1,541.61 240 \$0.00 ``` The asterisks (*) show you that the balances become equal somewhere between the 88th and 89th month. This would be somewhere between 7.33333333 years and 7.41666666667 years. The actual graph of the declining balance is not a straight line. It drops slowly in the early years and drops a lot faster in the later years. You can link to the following webiter and scroll down to the bottom and a graph of a remaining balance on a loan will show up. You can see that it is not a straight line. http://www.tvmcalcs.com/calculators/apps/excel_loan_amortization
 Polynomials-and-rational-expressions/384066: SIMPLIFY the expression: (1/x^2+6x+8)−(1/x^2−2x−8) and give your answer in the form of f(x)/g(x) My solution was (1/(x-4)(x+4)), but it was incorrect. 1 solutions Answer 271843 by Theo(3464)   on 2010-12-10 03:51:49 (Show Source): You can put this solution on YOUR website!The solution that I came up with is: I checked the solution out by assuming x was equal to 2 and solving both the original expression and the final expression. They came out to the same answer leading to the conclusion that I simplified correctly. My manual calculations are in the picture shown below: first I factored. then I found common denominator. then I combined numerator under same denominator. then I performed calculations on numerator. result was final answer.
 Pythagorean-theorem/384062: if the legs of a right angle are 9 and 17 what is the hypotenuse?1 solutions Answer 271840 by Theo(3464)   on 2010-12-10 03:27:45 (Show Source): You can put this solution on YOUR website!c = hypotenuse a = the shorter leg b = the longer leg formula is: c^2 = a^2 + b^2 becomes: c^2 = 9^2 + 17^2 = 370 c = sqrt(370) = 19.23538406
 logarithm/383497: A savings bond will pay \$5,000 at maturity 15 years from now. How much should you be willing to pay for the note now if money is worth 4.11% compounded semiannually?1 solutions Answer 271535 by Theo(3464)   on 2010-12-09 06:25:19 (Show Source): You can put this solution on YOUR website!will pay \$5000 in 15 years. interest rate per year is 4.11% compounded semiannual. formula to use is f = p * (1+i)^n your time periods need to be semi-years. because of that, you need to divide your interest rate by 2 and you need to multiply your years by 2. in your formula: f = 5000 p = what you want to find i = .0411 / 2 = .02055 n = 15 * 2 = 30 your formula becomes: 5000 = p * (1.02055)^30 your answer should be that you are willing to pay \$2,716.072716 for the savings bond. Let's see if that works. 1.02055^30 = 1.840893276 your formula becomes 5000 = p * 1.840893276 divide both sides of this equation by 1.840893276 and you get: p = 5000 / 1.840893276 = \$2,716.072716 We're good.
 Miscellaneous_Word_Problems/383501: I hope someone can help me. I am lost. I am to reduce the rational expression to the lowest terms. And assume that the variables represent only numbers which the denominators are nonzero. I do not undertand please to step by step. a^2-b^2/a-b Thank you so much!1 solutions Answer 271534 by Theo(3464)   on 2010-12-09 06:18:45 (Show Source): You can put this solution on YOUR website!problem is: (a^2 - b^2)/(a-b) (a^2 - b^2) can be factored to be (a-b) * (a+b) your expression becomes: ((a-b) * (a+b)) / (a-b) The (a-b) in the numerator and denominator cancel out and you are left with: 1 * (a+b) / 1 = (a+b) you had to know that (a^2 - b^2) is equivalent to (a+b) * (a-b). when you take (a+b) and multiply it by (a-b), you get: a^2 + ab - ab - b^2 which becomes a^2 - b^2 because the +ab and -ab cancel out.
 Miscellaneous_Word_Problems/383500: could someone help me work this out step by step so I can understand it. I am lost.I am suppose to build up the rational expression into an equivalent rational expression with the indicated denominator. Please help. Thank you -7yt/3x=?/18xyt 1 solutions Answer 271533 by Theo(3464)   on 2010-12-09 06:12:48 (Show Source): You can put this solution on YOUR website!-7yt/3x=?/18xyt it helps to separate what is being multiplied by each other by *. it also helps to surround operations that are going to be performed together by parentheses. your formula becomes: (-7*y*t) / (3*x) = ? / (18*x*y*t) multiply both sides of this equation by (18*x*y*t) you get: (18*x*y*t) * ((-7*y*t) / (3*x)) = ? since a * (b/c) = (a*b) / c, then your expression above is equivalent to: ((18*x*y*t) * (-7*y*t)) / (3*x) = ? (18*x*y*t) * (-7*y*t) is equal to (-126 * x * y^2 * t^2) this is because 18 * -7 = -126 and y * y = y^2 and t * t = t^2. (-126 * x * y^2 * t^3) / (3 * x) is equal to (-42 * y^2 * t^2) this is because -126/3 = -42 and x/x = 1 which is not shown but implied. your answer should be ? = (-42 * y^2 * t^2) you can confirm it by assigning values to each of the variables at random and seeing if the final expression is true after simplification. Assume x = 2, y = 3, t = 4 your original expression of (-7*y*t) / (3*x) = ? / (18*x*y*t) becomes (-7*y*t) / (3*x) = (-42 * y^2 * t^2) / (18*x*y*t) which becomes (-7*3*4) / (3*2) = (-42 * 3^2 * 4^2) / (18*2*3*4) this resolves to -84/6 = -6048 / 432 which resolves finally to -14 = -14 confirming that we did the calculations correctly.
 Age_Word_Problems/383490: Please help me solve this word problem: Formulas: y=be^rt; A=P(1+(r/n)^nt George has \$65 to invest. a. The bank gives him 8.2% continuous interest, how long will it take for George to accrue \$100? b. At 7.6% interest compound monthly, how much money will George have in 12 years?1 solutions Answer 271527 by Theo(3464)   on 2010-12-09 05:45:30 (Show Source): You can put this solution on YOUR website!y = b * e^rt sounds like this is the continuous compounding formula. george has \$65 to invest. at 8.2% continuous interest, how long will it take for george to accrue \$100? y = \$100 b = \$65 r = .082 t = time in years formula is \$100 = \$65 * e^(.082*t) divide both sides of this formula by 65 to get: 100/65 = e^(.082*t) take log of both sides to get: log(100/65) = log(e^(.082*t)) by laws of logarithms, log(x^y) = y*log(x). formula becomes log(100/65) = .082*t * log(e) divide both sides of equation by log(e) to get: log(100/65) / log(e) = .082*t divide both sides of this equation by .082 to get: t = (log(100/65) / log(e)) / .082 log(100/65) = .187086643 log(e) = log(2.718281828) = .434294482 log(100/65) / log(e) = .430782916 .430782916 / .082 = 5.253450196 you get t = (log(100/65) / log(e)) / .082 = 5.253450196. If we did this right, then t = 5.253450196 years your original formula is: 100 = 65 * e^(.082*t) this becomes: 100 = 65 * e^(.082*5.253450196) e represents the scientific constant of 2.718281828 100 = 65 * e^(.082*5.253450196) becomes: 100 = 65 * 2.718182818^(.082*5.253450196). Use your calculator to see that 100 = 100, making t = 5.253450196 correct. -------------------------------- second problem. At 7.6% interest compound monthly, how much money will George have in 12 years? Formula they show is: A=P(1+(r/n)^nt A = future value you want to find. P = \$65.00 r = .076 / 12 = .006333333 t = 12 * 12 = 144 the n they are showing is equal to 12. to compound monthly you take the number of years and multiply by 12 and you take the annual interest rate and divide it by 12. formula becomes: A = 65 * (1 + (.076/12))^(12*12) this becomes: A = 65 * (1.00633333)^144 = 161.3395912. In 12 years, george will have that much.
 Inequalities/383493: PLEASE HELP! I have solved this one but something looks wrong. Determine whether these numbers are solutions of the inequality : x-1 (greater then or equal to) 7; -5,1,4,20 I substituted -5,1,4,20 for x in the inequality and solved accordingly. I found there was NO solution for any of the numbers. Am I wrong and thank you.1 solutions Answer 271526 by Theo(3464)   on 2010-12-09 05:29:30 (Show Source): You can put this solution on YOUR website!if x is -5, 1, or 4, then x-1 >= 7 is false. if x is 20, then x-1 >= 7 is true, because 19 >= 7 -5-1 = -6 1-1 = 0 4-1 = 3 20-1 = 19
 Linear-equations/380971: Good Sunday Morning! I am trying to find the slope-intercept form of the equation of the line through the point (-1, 2), parallel to the line -4x- 9y = -7. Thanks for the help! Denise 1 solutions Answer 270346 by Theo(3464)   on 2010-12-05 09:36:43 (Show Source): You can put this solution on YOUR website!If the line is parallel then it has the same slope. The line it is parallel to is the line given by the equation -4x - 9y = -7 convert that line to the slope intercept form by solving for y. you will get y = (-4/9)*x + (7/9) The slope of that line is equal to (-4/9) That will be the slope of your new line. The general equation for the slope intercept form is y = mx + b where m is the slope and b is the y intercept. Since you have the slope, the general form becomes y = (-4/9)*x + b To find the y intercept, you substitute the point values that the line passes through. The point values that you have that the line passes through are (-1,2). That means that x = -1 and y = 2. Substitute in the equation of y = (-4/9)*x + b to get: 2 = (-4/9)*(-1) * b Simplify to get: 2 = (4/9) * b Solve for b to get: b = 2 - (4/9) = (18/9) - (4/9) = (14/9) The equation of your parallel line should be: y = (-4/9)*x + (14/9) graph both equations and you will see that they are parallel. To prove that the slope intercept form of your original equation is good, you need to substitute the (x,y) values derived from the slope intercept form of the equation and see if the original equation is true. Your original equation is -4x-9y=-7. The slope intercept form of this equation is y = (-4/9)*x + (7/9) If we let x = 18, then y = (-4/9)*18 + (7/9) = -7.22222222 If we let x = 18 and y = -7.22222222, then -4x - 9y = -7 becomes -72 + 65 = -7 the translation from the standard form of the equation (-4x-9y=-7) to the slope intercept form of the equation ( y = (-4/9)*x + (7/9)) is good. The line parallel to it is therefore good because we calculated that correctly based on the slope intercept form of the original equation.
 Evaluation_Word_Problems/380967: A student receives his grade report from a local community college, but the GPA is smudged. He took the following classes: a 2 hour credit art, a 3 hour credit history, a 4 hour credit science course, a 3 hour credit mathematics course, and a 1 hour science lab. He received a B in the art class, an A in the history class, a C in the science class, a B in the math class, and an A in the science lab. What was his GPA if the letter grades are based on a 4 point scale? (A = 4, B = 3, C = 2, D = 1, F = 0)1 solutions Answer 270345 by Theo(3464)   on 2010-12-05 09:08:17 (Show Source): You can put this solution on YOUR website!A = 4, B = 3, C = 2, D = 1, F = 0 The following chart shows you how the calculation is set up. ```Class Hours Grade Grade Points Grade Point Hours Art 2 B 3 6 History 3 A 4 12 Science 4 C 2 8 Mathematics 3 B 3 9 Science Lab 1 A 4 4 ``` A student receives his grade report from a local community college, but the GPA is smudged. He took the following classes: a 2 hour credit art, a 3 hour credit history, a 4 hour credit science course, a 3 hour credit mathematics course, and a 1 hour science lab. He received a B in the art class, an A in the history class, a C in the science class, a B in the math class, and an A in the science lab. What was his GPA if the letter grades are based on a 4 point scale? (A = 4, B = 3, C = 2, D = 1, F = 0) The total grade point hours are equal to 39 The total hours are equal to 13 The grade point average = 39/13 = 3 This equates to an overall average of B. You multiply the grade points by the hours to get the grade point hours. The total grade point hours divided by the total hours equals the grade point average.
 Triangles/380867: hey i would like to know how to find the area of a geometric figure 10 yd 13yd1 solutions Answer 270342 by Theo(3464)   on 2010-12-05 08:38:19 (Show Source): You can put this solution on YOUR website!I presume the figure is a rectangle? It has length of 10 yards and a width of 13 yards? The area would be length * width = 10 * 13 = 130 square yards. You posted this under triangles? If this figure is a triangle, then we need to make some assumptions. The area of a triangle is 1/2 * b * h where b is the base and h is the height. If you assume that the base is 10 yards and the height is 13 yards, then the area would be 1/2 * 10 * 13 = 65 square yards. If you assume that the base is 13 yards and the height is 10 yards, you would get the same area equal to 1/2 * 13 * 10 = 65 square yards.
 Polynomials-and-rational-expressions/380855: Worker efficiency. In a study of worker efficiency at Wong Laboratories it was found that the number of components assembled per hour by the average worker t hours after starting work could be modeled by the formula N(t) = 3t^3 + 23t^2 + 8t a) Rewrite the formula by factoring the right-hand side completely. b) Use the factored version of the formula to find N(3). c) What is the time at which the workers are most efficient. d) What is the maximum number of components assembled per hour during an 8-hour shift.1 solutions Answer 270341 by Theo(3464)   on 2010-12-05 08:31:26 (Show Source): You can put this solution on YOUR website!the formula you gave is N(t) = 3t^3 + 23t^2 + 8t I'll make x = t so this can be graphed and referenced easily. the equation becomes n(x) = 3x^3 + 23x^2 + 8x the graph of this formula is shown below: the formula as given doesn't make any sense. there is no maximum when x is greater than 0. x cannot be less than 0 so any values on the graph where x is less than 0 are to be ignored. your graph would make more sense if the equation was: N(t) = -3t^3 + 23t^2 + 8t which I would translate to: n(x) = -3x^3 + 23x^2 + 8x. that graph is shown below: now you have a maximum when x is greater than 0 which is more realistic. I'll solve for n(x) = -3x^3 + 23x^2 + 8x. x can be factored out to get: n(x) = x * (-3x^2 + 23x + 8) (-3x^2 + 23x + 8) factors out to be (3x+1) * (-x+8) the completely factored equation becomes: n(x) = x * (3x+1) * (-x+8) when x = 0, n(x) = 0 as it should because no time worked = no components produced. when x = 3, n(x) = (3) * (10) * (5) = 150 using the factored equation. when x = 3, n(x) = -3(3^3) + 23*(3^2) + 8*3 = -81 + 207 + 24 = 150 you get the same answer with the factored equation and the non factored equation which confirms that the factorization is good. I believe we need to use calculus to find the maximum / minimum point of this cubic equation. I couldn't find any formula not using calculus on the web, so we'll use calculus. The maximum point occurs when the derivative of the equation equals 0. the equation is -3x^3 + 23x^2 + 8x The derivative of this equation is -9x^2 + 46x + 8 This derivative is 0 when x = 5.279477927 That could not be factored easily, so I used the quadratic formula of: x = (-b +/- sqrt(b^2-4ac))/(2a) When x = 5.279477927, n(x) = 241.8493507 I'll add a line at n(x) = 241.8493507 to the graph of the equation and it should confirm that is the maximum point. the graph looks like this: The graph confirms the maximum point. Since x represents t which represents the number of hours after starting work, then the maximum amount of output per hour occurs 5.279477927 hours after work starts. The maximum amount of output per hour during the 8 hour shift becomes 241.8493507 units per hour which occurs at that time. Best I can do in the short period of time allotted to the solving of this problem. I'm not a calculus guru, but I know a little bit about it, and the little bit I know was helpful in this case. If you were not to use calculus, then I don't really know how to find the maximum point of the cubic equation other than by iteration.
 Linear_Equations_And_Systems_Word_Problems/380854: When soft drinks sold for \$0.80 per can at football games, approximately 6500 cans were sold. When the price was raised to \$1.00 a can, the demand dropped to 4000. Assume that the relationship between the price p and the demand y is linear. Write a linear function giving the demand y as a function of p.1 solutions Answer 270337 by Theo(3464)   on 2010-12-05 07:12:17 (Show Source): You can put this solution on YOUR website!let y = number of cans sold. let x = price per can. when x = .8, y = 6500 when x = 1, y = 4000 you have 2 points from which you can generate a linear equation. the point slope form of the linear equation is y = mx + b. m is the slope and b is the y intercept. the general form of the slope of the line is (y2-y1)/(x2-x1) you have 2 points. they are: (.8,6500) and (1,4000) x1 = .8 y1 = 6500 x2 = 1 y2 = 4000 your slope is (y2-y1)/(x2-x1) which becomes (4000-6500) / (1-.8). this becomes -2500 / .2 which becomes -12500. that's your slope and the slope intercept form of your equation becomes: y = -12500*x + b to find the y intercept, substitute one of the points for y and x. we'll use (.8,6500) the equation becomes: 6500 = -12500*(.8) + b that becomes 6500 = -10000 + b solve for b to get b = 16500 your equation becomes: y = -12500*x + 16500 when x = .8, y becomes 6500 when x = 1, y becomes 4000 The slope is -12500 the y intercept is 16500 the equation is y = -12500*x + 16500 graph this equation and it looks like this: when x = 0, y = 16500 when x = .8, y = 6500 when x = 1, y = 4000 It's hard to see from the graph, but if you plot real careful the intersection of x = 1 and y = 4000, you will see that they intersect on the line of y = -12500*x + 16500. The more exact way is to simply solve the equation of y = -12500*x + 16500 when x = .8 you get y = -12500*(.8) + 16500 which becomes y = -10000 + 16500 which becomes y = 6500.
 Money_Word_Problems/380958: A retailer purchased 70 reams of typewriter paper for \$9,730. At the end of the week, he found that there are still 13 reams left, the cost of which was \$1,807. Find the cost of goods sold.1 solutions Answer 270336 by Theo(3464)   on 2010-12-05 06:44:11 (Show Source): You can put this solution on YOUR website!he bought 70 for 9730 at a cost of 9730/70 = 139 apiece. he had 13 left at a cost of 1807. cost of goods sold is 9730 - 1807 = 7923 number of goods sold is 70 - 13 = 57 57 * 139 = 7923 number confirm each other. 7923 is your answer.
 Angles/380897: if m<1 is 75 what is the measure of its complementary angle1 solutions Answer 270334 by Theo(3464)   on 2010-12-05 06:41:36 (Show Source): You can put this solution on YOUR website!the measure of an angle plus its complement is equal to 90 degrees. if the angle is 75 degrees, then its complement must be equal to 15 degrees because 75 + 15 = 90.
 Exponential-and-logarithmic-functions/380911: The function f(x)=9x+1 is one-to-one. Find the inverse1 solutions Answer 270333 by Theo(3464)   on 2010-12-05 06:38:53 (Show Source): You can put this solution on YOUR website!f(x) = 9x + 1 let y = f(x) you get y = 9x + 1 let y = x and x = y you get x = 9y + 1 solve for y to get y = (x-1)/9 that's your inverse function. if it is a true inverse function, it will be a reflection of the original function about the line y = x. to show that, then graph the functions y = x, y = 9x+1, y = (x-1)/9. that graph is shown below: you can pretty much eyeball it and see that the graph of y = (x-1)/9 is a reflection of y = 9x+1 about the line y = x. a more exact interpretation is that f(x,y) in the normal equation should be equal to f(y,x) in the inverse equation. how you find that out is as follows: let x = 5 in the normal equation. then you get y = 9x + 1 which becomes 46 in your normal equation, when x = 5, then y = 46 in your inverse equation, when x = 46, you should get y = 5. the y in the normal equation becomes the x in the inverse equation. the x in the normal equation becomes the y in the inverse equation. when x = 46, y = (x-1)/9 becomes y = (46-1)/9 becomes y = 45/9 becomes 5. f(x,y) = f(5,46) in your normal equation. f(y,x) = f(46,5) in your inverse equation. this proves they are inverse equations. the inverse equation undoes what the normal equation does. normal equation takes 5 and makes it 46. inverse equation takes 46 and makes it 5.
 Exponential-and-logarithmic-functions/380916: write the expression (log base of b)(2y+5)-4(log base of b)(y+3) as a single logarithm1 solutions Answer 270329 by Theo(3464)   on 2010-12-05 06:28:20 (Show Source): You can put this solution on YOUR website!that expression would be shown as: log(b,(2y+5)) - 4*log(b,(y+3)) in general x * log(y) = log(y^x) your expression becomes: log(b,(2y+5)) - log(b,((y+3)^4)) in general log(x) - log(y) = log(x/y) your expression becomes: log(b,((2y+5)/(y+3)^4)) to show you how this works, we will let b = 10 because your calculator can do logs to the base of 10 (usually called the LOG function). we will let y = 5 (chosen at random small enough to calculate easily). your original expression becomes: log(2y+5) - 4*log(y+3) the base of 10 is implied. substituting 5 for y, we get: log(2*5+5) - 4*log(5+3) which becomes: log(15) - 4*log(8) which becomes -2.436268689 looking at our final expression of: log(b,(2y+5)/(y+3)^4), we get: log((2y+5)/(y+3)^4). the base of 10 is implied. substituting 5 for y, we get: log(15/8^4) which becomes log(15/4096). using our calculator, we get log(15/4096) = log(.003662109) which equals -2.436268689 we get the same answer either way, so the translation is good, and the answer to your question is: log(b,2y+5) - 4*log(b,y+3) = log(b,(2y+5)/(y+3)^4) which looks like:
 Exponential-and-logarithmic-functions/380915: Write the expression log2(6x/y) as a sum and/ or difference of logarithms1 solutions Answer 270327 by Theo(3464)   on 2010-12-05 06:10:44 (Show Source): You can put this solution on YOUR website!log(2,6x/y) is equal to the log of 6x/y to the base of 2. it means the same as what you wrote as log2(6x/y). the notation, however, is more in line with the way the algebra.com formula generators work. using the algebra.com formula generator, log(2,6x/y) will show up as All you do is put 3 { in front of it and 3 } behind it. in general, log(x/y) = log(x) - log(y) your equation becomes: log(2,6x/y) = log(2,6x) - log(2,y) in general, log (x*y) = log(x) + log(y) your equation becomes: log(2,6x/y) = log(2,6) + log(2,x) - log(y) the concept is the same regardless of the base. if the base were 10, then it would be shown as: log(10,6x/y) = log(10,6) + log(10,x) - log(10,y) to show you how it works, we'll use log to the base 10 because your calculator can handle that. also, log(10,x) is normally shown as log(x). the base of 10 is implied. let's take log(6*15/30) this should be translated to log(6) + log(15) - log(30) which is the same treatment we provided above. using our calculator, we get log(6*15/30) = log(3) = .477121255 using our calculator again, we get log(6) + log(15) - log(30) = : .77815125 + 1.176091259 - 1.477121255 = .477121255 we get the same answer, as we should. same concepts works with any base, so the answer to your question is: log(2,6x/y) = log(2,6) + log(2,x) - log(2,y).
 Exponential-and-logarithmic-functions/380907: For f(x)=8x+6 and g(x)= x squared find (g of f)(x)1 solutions Answer 270326 by Theo(3464)   on 2010-12-05 05:57:20 (Show Source): You can put this solution on YOUR website!you are given: f(x) = 8x+6 g(x) = x^2 you want to find: g(f(x)) start with g(x) = x^2 replace x with f(x) to get g(f(x)) = (f(x))^2 replace f(x) with 8x+6 to get g(8x+6) = (8x+6)^2 solve by squaring 8x+6 to get g(f(x)) = 64x^2 + 96x + 36
 Probability-and-statistics/380950: Investing is a game of chance. Suppose there is a 39% chance that a risky stock investment will end up in a total loss of your investment. Because the rewards are so high, you decide to invest in five independent risky stocks. Find the probability that at least one of your five investments becomes a total loss. I'm so confused! Thank you so much for your help! 1 solutions Answer 270309 by Theo(3464)   on 2010-12-05 04:39:25 (Show Source): You can put this solution on YOUR website!probability of total loss of one risky investment = .39 you take 5 risky investments, each with a probability of total loss of .39. since the probability of total loss of each investment is .39, then the probability that the investment will not be a total loss is 1 - .39 = .61 the probability that none of the 5 investments will be a total loss is .61^5 = .08445963. the probability that at least one of these investments will be a total loss is 1 - .61^5. that probability would be .91554037
 Equations/380936: How do I covert 1256kg to grams? I came up with 1256000 Is this correct? Thank you!1 solutions Answer 270306 by Theo(3464)   on 2010-12-05 04:16:53 (Show Source): You can put this solution on YOUR website!to find the conversion factors, it's convenient to go to www.google.com and put in the conversion you are looking for. In your case, you want to convert from kilograms to grams. The google search string entered would be: kilograms to grams. the answer returned would be: 1 kilogram = 1000 grams If you want to convert from kilograms to grams, you would have to multiply the number of kilograms by 1000. If you start with 1256 kg, then your answer would bge 1,256,000 grams. You did good.