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# Recent problems solved by 'nerdybill'

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 Miscellaneous_Word_Problems/212467: Let f(t)=t^2+4t+2. Find a value of t such that the average rate of change of f(t) from 0 to t equals 13 1 solutions Answer 160609 by nerdybill(6958)   on 2009-09-10 23:17:15 (Show Source): You can put this solution on YOUR website!Let f(t)=t^2+4t+2. Find a value of t such that the average rate of change of f(t) from 0 to t equals 13 . "average rate of change" is simply the instantaneous slope when t=13. . To find it, you can find the derivative of f(t): f(t)=t^2+4t+2 f'(t)=2t+4 . Now, find f'(13): f'(t)=2t+4 f'(13)=2(13)+4 f'(13)=26+4 f'(13)=30
 Exponents-negative-and-fractional/212536: Can you help me solve this equation? I have tried to solve it for two days now and still can't get it right. 1 solutions Answer 160495 by nerdybill(6958)   on 2009-09-10 16:58:09 (Show Source): You can put this solution on YOUR website!Starting with: . . . Multiply by : . * . . Doing so yields: . . Now, factoring the denominator: . Canceling like-terms we end up with:
 Miscellaneous_Word_Problems/212103: A soccer ball is kicked straight up with an initial velocity of 32 ft. per second. Its height above the earth is given by s (t) = -16t2 + 32t, where,‘s’ is the height in ft. at any time, ‘t’ seconds. What is the maximum height reached by the ball? How long after it was thrown up the ball reaches its maximum height? 1 solutions Answer 160199 by nerdybill(6958)   on 2009-09-09 09:47:29 (Show Source): You can put this solution on YOUR website!A soccer ball is kicked straight up with an initial velocity of 32 ft. per second. Its height above the earth is given by s (t) = -16t2 + 32t, where,‘s’ is the height in ft. at any time, ‘t’ seconds. What is the maximum height reached by the ball? How long after it was thrown up the ball reaches its maximum height? . The "vertex" of the parabola will give you both. . First, find the "axis of symmetry": t = -b/2a t = -32/2(-16) t = -32/-32 t = 1 second (answer to second question) . The "height" is found by plugging the value above back into: s(t) = -16t^2 + 32t s(1) = -16(1)^2 + 32(1) s(1) = -16 + 32 s(1) = 16 feet (answer to first question)
 Miscellaneous_Word_Problems/212104: What is the length of the diagonal of a rectangular bill board with sides of lengths 5 ft and 12 ft.1 solutions Answer 160194 by nerdybill(6958)   on 2009-09-09 09:03:44 (Show Source): You can put this solution on YOUR website! What is the length of the diagonal of a rectangular bill board with sides of lengths 5 ft and 12 ft. . You need to apply Pythagorean theorem: a^2 + b^2 = c^2 where a,b are the sides of the right triangle and c is the hypotenuse . 5^2 + 12^2 = c^2 25 + 144 = c^2 169 = c^2 13 ft = c (hypotenuse)
 test/204550: Hi there, I have an exam today and I was doing a practise exam where I came across this question and was unable to solve it. Could one of the tutors please help me with it, I'd be more than grateful!!!! Scientists have 50g of a radioactive element that decomposes with a half-life of 15 seconds. (3 marks:1 mark each) a) Write an equation that predicts the amount of the element remaining as a function of time. b) How much of the element would remain after 2 minutes? c) How long would the element take to decompose to the point where there is only 0.01g remaining?1 solutions Answer 160189 by nerdybill(6958)   on 2009-09-09 08:50:48 (Show Source): You can put this solution on YOUR website!Scientists have 50g of a radioactive element that decomposes with a half-life of 15 seconds. (3 marks:1 mark each) a) Write an equation that predicts the amount of the element remaining as a function of time. P = Ce^(kt) 25 = 50e^(k*15) 25/50 = e^(k*15) 1/2 = e^(k*15) ln(1/2) = 15k ln(1/2)/15 = k Your equation then: P = 50e^[(ln(1/2)/15)t] . b) How much of the element would remain after 2 minutes? 2 minutes = 2*60 = 120 secs plug it into the equation: P = 50e^[(ln(1/2)/15)t] P = 50e^[(ln(1/2)/15)120] P = 50(.00390625) P = .195 grams . c) How long would the element take to decompose to the point where there is only 0.01g remaining? 0.01 = 50e^[(ln(1/2)/15)t] 0.0002 = e^[(ln(1/2)/15)t] -8.5172 = (ln(1/2)/15)t -8.5172/(ln(1/2)/15) = t 184.32 secs = t
 Linear-equations/212098: This question is from textbook Geometry How would you solve: -3x+y=7 3x+2y=2 using elimination1 solutions Answer 160180 by nerdybill(6958)   on 2009-09-09 06:03:14 (Show Source): You can put this solution on YOUR website!Add the two equations together: -3x+y=7 3x+2y=2 -------- 0x+3y= 9 3y = 9 y = 3 . Substitute the above into equation 1 and solve for x: -3x+y=7 -3x+3=7 -3x=4 x = -4/3 . Solution: x = -4/3 y = 3 Or, (x,y) = (-4/3, 3)
 Numeric_Fractions/212099: This question is from textbook Beginning Algebra Ahmad Wishes to purchase mulch to cover his garden. The garden measures 63/8 ft by 81/8 ft. He wants the mulch to be 1/3 ft deep. How much mulch should Ahamad order if he must order a whole number of cubic feet? My try: I think th ethree frations should be multiplied: (63/8)(81/8)(1/3)= 5103/1921 solutions Answer 160179 by nerdybill(6958)   on 2009-09-09 06:00:22 (Show Source): You can put this solution on YOUR website!Ahmad Wishes to purchase mulch to cover his garden. The garden measures 63/8 ft by 81/8 ft. He wants the mulch to be 1/3 ft deep. How much mulch should Ahamad order if he must order a whole number of cubic feet? My try: I think th ethree frations should be multiplied: (63/8)(81/8)(1/3)= 5103/192 . You would be correct. However, the part of the problem which states: "he must order a whole number of cubic feet" Now, you must convert 5103/192 to a whole number 5103/192 = 26.578125 = 27 cubic feet
 Miscellaneous_Word_Problems/211874: This question is from textbook just-in-time algebra and trigonometry a 2 ft. wire is cut into two pieces. one of the pieces, of length x, is bent into a circle. the other piece is bent into a rectangle whose length is twice the size of its width. what is the total area of the two shapes as a function of x.1 solutions Answer 160080 by nerdybill(6958)   on 2009-09-08 16:01:52 (Show Source): You can put this solution on YOUR website! a 2 ft. wire is cut into two pieces. one of the pieces, of length x, is bent into a circle. the other piece is bent into a rectangle whose length is twice the size of its width. what is the total area of the two shapes as a function of x. . Let x = circumference of circle then 2-x = perimeter of rectangle . Since for any circle: circumference = (pi)d = 2(pi)r then x = 2(pi)r r = x/(2(pi)) . Area of circle: (pi)r^2 =(pi)[x/(2(pi))]^2] =(pi)[x^2/(4(pi)^2)] =x^2/(4(pi)) . rectangle: Let w = width then 2w = length perimeter = 2(w + 2w) perimeter = 2(3w) 2-x = 2(3w) 2-x = 6w (2-x)/6 = w (width) . Length: 2w = 2[(2-x)/6] = (2-x)/3 (length) . Area of rectangle: w*L = (2-x)/6 * (2-x)/3 = (2-x)^2/18 . Total area: "area of circle" + "area of rectangle" = x^2/(4(pi)) + (2-x)^2/18 = 18x^2/(72(pi)) + (4(pi))(2-x)^2/(72(pi)) = [18x^2 + (4(pi))(2-x)^2]/(72(pi)) = [18x^2 + (4(pi))(2-x)(2-x)]/(72(pi)) = [18x^2 + (4(pi))(4-4x+x^2)]/(72(pi)) = [18x^2 + 16(pi) - 16(pi)x + 4(pi)x^2]/(72(pi)) = [18x^2 + 4(pi)x^2 + 16(pi) - 16(pi)x]/(72(pi)) = [2x^2(9 + 2(pi)) + 16(pi)(1-x)]/(72(pi)) = 2[x^2(9 + 2(pi)) + 8(pi)(1-x)]/(72(pi)) = [x^2(9 + 2(pi)) + 8(pi)(1-x)]/(36(pi)) = [x^2(9 + 2(pi)) + 8(pi)(1-x)]/(36(pi))
 Miscellaneous_Word_Problems/211670: The monthly profit P (in dollars) that Jim makes on the sale of x mobile homes is determined by the formula P = x^2 + 5x – 50. Find the values of x for which Jim makes profit. 1 solutions Answer 159968 by nerdybill(6958)   on 2009-09-07 22:36:29 (Show Source): You can put this solution on YOUR website!P = x^2 + 5x – 50 . Find out the times when profit is zero: P = x^2 + 5x – 50 0 = x^2 + 5x – 50 0 = (x+10)(x-5) x = {-10, 5} You can throw out the negative answer (you can't sell negative mobile homes) leaving you with: x = 5 . Therefore, x > 5 is your answer
 Equations/211719: How many square yards of carpet are needed to carpet a room that is 15 ft by 25 ft?1 solutions Answer 159967 by nerdybill(6958)   on 2009-09-07 22:32:25 (Show Source): You can put this solution on YOUR website!Area of a room that is: 15 ft by 25 ft is 15 * 25 = 375 square feet . 1 square yard = 9 square feet . Therefore, 375/9 = 41.67 square yards (or, 41 and 2/3 square yards)
 Average/211691: The results of a survey for an airline are shown below Traveler Male Female Total Business 47 72 119 Vacation 71 64 135 Total 118 136 254 Use the chart to find the probability that the traveler was a) male b) on vacation given the traveler was female c) male given the traveler was on vacation 1 solutions Answer 159951 by nerdybill(6958)   on 2009-09-07 21:45:41 (Show Source): You can put this solution on YOUR website!The results of a survey for an airline are shown below Traveler Male Female Total Business 47 72 119 Vacation 71 64 135 Total 118 136 254 Use the chart to find the probability that the traveler was a) male "males on business or vacation"/"total travelers" (47+71)/(119+135) 118/254 * 100 .465 * 100 46.5% . b) on vacation given the traveler was female "female traveling on vacation"/"total female travelers" 64/(72+64) * 100 = 64/136 * 100 = 47.1% . c) male given the traveler was on vacation "males traveling on vacation"/"total travelers on vacation" 71/(71+64) * 100 = 71/135 * 100 = 52.6%
 Linear-equations/211709: This question is from textbook intermediate algebra determine the y- intercept. g(x)=-4x-31 solutions Answer 159947 by nerdybill(6958)   on 2009-09-07 21:37:33 (Show Source): You can put this solution on YOUR website!y = mx + b is the "slope-intercept" form of a straight line m is slope b is the y-intercept at (0,b) . Since your equation: g(x)=-4x-3 is already in "slope-intercept" form -- by inspection we have y-intercept is at (0,-3)
 logarithm/211484: how do you solve... logx 16 = -41 solutions Answer 159820 by nerdybill(6958)   on 2009-09-07 12:43:30 (Show Source): You can put this solution on YOUR website!Applying "log rules": logx 16 = -4 16 = x^(-4) 16 = 1/x^4 16x^4 = 1 x^4 = 1/16 x = 1/2
 Miscellaneous_Word_Problems/211475: This question is from textbook College Algebra Equation,Inequalities, and mathematical modeling Renting an Apartment: Three students are planning to rent an apartment for a year and share equally in the cost. By adding a fourth person, each person could save \$75 a month. How much is the monthly rent?1 solutions Answer 159819 by nerdybill(6958)   on 2009-09-07 12:40:28 (Show Source): You can put this solution on YOUR website!Renting an Apartment: Three students are planning to rent an apartment for a year and share equally in the cost. By adding a fourth person, each person could save \$75 a month. How much is the monthly rent? . Let x=rent per student if 3 room mates then 12(3x) = "total rent for 3 room mates (in a year)" . If four students rent 12(4(x-75)) = "total rent for 4 room mates" . set them equal and solve for x: 12(3x) = 12(4(x-75)) 3x = 4(x-75) 3x = 4x-300 300+3x = 4x \$300 per month = x . So, that's \$300 per month for each room mate (if there are only 3 of them) . Monthly rent = 3(300) = \$900 per month
Rectangles/211479: This question is from textbook Beginning & Intermediate Algebra
An object is thrown upward from the top of an 80 foot building with an initial velocity of 64 feet per second. The height h of an object after t seconds is given by the quadratic equation (((h=-16t^2+64t+80))). When will the object hit the ground.
1 solutions

Answer 159818 by nerdybill(6958)   on 2009-09-07 12:33:39 (Show Source):
You can put this solution on YOUR website!
The problem gives you:
h=-16t^2+64t+80
where
h is height in feet
t is time in secs
.
If it hits the ground, h = 0 so
set h=0 and solve for t:
h=-16t^2+64t+80
0=-16t^2+64t+80
0=-4t^2+16t+5
Solving using the quadratic equation yields:
t = {-0.29, 4.29}
We can throw out the negative solution leaving
t = 4.29 seconds
.
 Solved by pluggable solver: SOLVE quadratic equation with variable Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=336 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: -0.29128784747792, 4.29128784747792. Here's your graph:

 Miscellaneous_Word_Problems/211435: Randy has a jar containing 63 coins, which are either quarters or nickels. The total value of the jar is \$6.75. How many of each type of coin does he have?1 solutions Answer 159787 by nerdybill(6958)   on 2009-09-07 09:01:02 (Show Source): You can put this solution on YOUR website!Randy has a jar containing 63 coins, which are either quarters or nickels. The total value of the jar is \$6.75. How many of each type of coin does he have? . Let q = number of quarters then 63-q = number of nickels . .25q + .05(63-q) = 6.75 .25q + 3.15 - .05q = 6.75 .20q + 3.15 = 6.75 .20q = 3.60 q = 3.60/.20 q = 18 (number of quarters) . Number of nickels: 63-q = 63-18 = 45
 Miscellaneous_Word_Problems/211424: A 45 foot rope is to be cut into 3 pieces. The second piece must be twice as long as the first piece and the third piece must be 9 feet longer than 3 times the length of the second piece. How long should each of the three pieces be? I know: x= the 1st piece 2x= the second piece1 solutions Answer 159782 by nerdybill(6958)   on 2009-09-07 02:07:29 (Show Source): You can put this solution on YOUR website!A 45 foot rope is to be cut into 3 pieces. The second piece must be twice as long as the first piece and the third piece must be 9 feet longer than 3 times the length of the second piece. How long should each of the three pieces be? . Good start: x= the 1st piece 2x= the second piece From: "the third piece must be 9 feet longer than 3 times the length of the second piece" we have: 3(2x)+9 = the third piece . x + 2x + 3(2x)+9 = 45 x + 2x + 6x + 9 = 45 9x + 9 = 45 9x = 36 x = 4 feet (1st piece) . 2x= 2*4= 8 feet (2nd piece) . 3(2x)+9 = 3(8)+9 = 24+9 = 33 feet (3rd piece)
 Polynomials-and-rational-expressions/211419: This question is from textbook okay i've tried but i just cant seem to get the hang of this... how do i solve this equation? 6x^3+24x1 solutions Answer 159774 by nerdybill(6958)   on 2009-09-07 00:31:38 (Show Source): You can put this solution on YOUR website!You can't really "solve" it. You can, however, factor it. . 6x^3+24x = 6(x^3+4x) = 6x(x^2+4)
 Probability-and-statistics/211223: Someone Please Help, with this solution A card is selected from a deck of 52 playing cards. Find the probability of selecting A) a seven given the card is not a face card (An ace is not a face card) B) a spade given the card is red1 solutions Answer 159577 by nerdybill(6958)   on 2009-09-06 08:51:04 (Show Source): You can put this solution on YOUR website!A card is selected from a deck of 52 playing cards. Find the probability of selecting A) a seven given the card is not a face card (An ace is not a face card) Since ALL "seven" cards are NOT face cards, probability is 4/52 = 0.077 = 7.7% . B) a spade given the card is red Since ANY spade is not red: 13/52 = 0.25 = 25%
 Miscellaneous_Word_Problems/211170: Ths is a summer math problem I was assigned to complete. A punter can kick a football with an initial velocity of 48 feet per second. How many seconds will it take for the ball to return to the ground?(Hint: Use the formula h=vt-16t2*)I was unsure whether or not this formula was correct for some reason I thought the formula was h=v-16t2*. I still need help wth completing the problem though. Thank you in adance the sooner this is answered the better. *The 2 in the problems above is a square.1 solutions Answer 159532 by nerdybill(6958)   on 2009-09-05 19:09:23 (Show Source): You can put this solution on YOUR website!A punter can kick a football with an initial velocity of 48 feet per second. How many seconds will it take for the ball to return to the ground?(Hint: Use the formula h=vt-16t2*) . No, h=vt-16t^2 is correct since v (velocity) is feet/sec and if you didn't multiply with time your units would not work. . Since h = height (in feet) set to zero to find out when it hits the ground . 0 = vt-16t^2 plugging in the given initial velocity: 0 = 48t-16t^2 0 = 3t-t^2 0 = t(3-t) t = {0, 3} . Well, 0 secs is "before" the punter kicked the ball so we're left with: t = 3 seconds
 Functions/211168: What is the domain of (x-5)/(x^2+5x+6)1 solutions Answer 159517 by nerdybill(6958)   on 2009-09-05 17:42:26 (Show Source): You can put this solution on YOUR website!Domain specifies the values of 'x' can take. In this case, you just want to make sure the denominator does NOT equal zero. . 0 = (x^2+5x+6) 0 = (x+2)(x+3) x = {-2,-3} . Therefore, the domain is all real numbers except -2 and -3. . Or, you can write as: (-oo, -3) U (-2, -3) U (-3, +oo) where oo is the infinity symbol
 Graphs/211166: This question is from textbook Determine whether each pair of line is parallel, perpendicular, or neithr 2x-5y=7 an 15y-5=6x1 solutions Answer 159516 by nerdybill(6958)   on 2009-09-05 17:38:37 (Show Source): You can put this solution on YOUR website!Strategy: Put both equation into "point-slope" form to compare slopes: y = mx + b where m is the slope . 2x-5y=7 -5y=-2x+7 y = (2/5)x - 7/5 and 15y-5=6x 15y=6x+5 y=(6/15)x+5/15 y=(2/5)x+5/15 . Since both lines have the same slope, the lines are parallel.
 Exponents-negative-and-fractional/211086: This question is from textbook If anyonw could help I would appreciate it. Rewrite the following using only positive exponents (2/5)^-2 and 5(x^2)^-11 solutions Answer 159487 by nerdybill(6958)   on 2009-09-05 06:10:52 (Show Source): You can put this solution on YOUR website! = . = =
 Equations/211077: Solve. Where appropriate, include approximations to the nearest thousandth. If no solution exists, state this. log x = 2.5 I came up with the answer below. Is this correct? logx=2.5 10^(logx)=10^(2.5) x=10^(2.5) x=316.231 solutions Answer 159480 by nerdybill(6958)   on 2009-09-05 01:48:37 (Show Source): You can put this solution on YOUR website!Yes, you are correct. To check, you could simply plug your answer back into: log x = 2.5
 Quadratic_Equations/210919: Solve the equation by introducing a substituition that transforms this equation to quadratic form. 3=1/(x+1)^2+2/(x+1) 1 solutions Answer 159326 by nerdybill(6958)   on 2009-09-04 02:19:05 (Show Source): You can put this solution on YOUR website!3=1/(x+1)^2+2/(x+1) . Let t = 1/(x+1) then, substitute the above in to the original: 3 = t^2 + 2t 0 = t^2 + 2t - 3 0 = (t+3)(t-1) t = {-3,1} . But, remember t = 1/(x+1) -3 = 1/(x+1) -3(x+1) = 1 -3x-3 = 1 -3x = 4 x = -4/3 . 1 = 1/(x+1) (x+1) = 1 x = 0 . x = {-4/3, 0}
 Quadratic_Equations/210917: Solve the equation by introducing a substituition that transforms this equation to quadratic form. x^-8-17x^-4+16=01 solutions Answer 159325 by nerdybill(6958)   on 2009-09-04 02:14:21 (Show Source): You can put this solution on YOUR website!x^-8-17x^-4+16=0 . Let t = x^-4 now, substitute the above into the original: t^2 - 17t + 16 = 0 (t-16)(t-1) = 0 t = {1, 16} . To find x, use the fact that t=x^-4 1 = x^-4 1 = 1/x^4 x = 1 . 16 = x^-4 16 = 1/x^4 x^4 = 1/16 x = 1/2 . x = +-{1, 1/2}
 Travel_Word_Problems/210906: This question is from textbook A boat travels due east for a distance of 15 miles. It then travels due North for a distance of 20 miles, at which point it drops anchor. How many miles is the boat from it's starting point?1 solutions Answer 159321 by nerdybill(6958)   on 2009-09-04 00:15:07 (Show Source): You can put this solution on YOUR website! A boat travels due east for a distance of 15 miles. It then travels due North for a distance of 20 miles, at which point it drops anchor. How many miles is the boat from it's starting point? . Applying Pythagorean theorem: Let d = distance then d^2 = 15^2 + 20^2 d^2 = 225 + 400 d^2 = 625 d = sqrt(625) d = 25 miles
 Geometry_Word_Problems/210902: Solve 8x^2y+3y=2x for y as a function of x (express in calculator syntax form)1 solutions Answer 159316 by nerdybill(6958)   on 2009-09-03 23:25:51 (Show Source): You can put this solution on YOUR website!8x^2y+3y = 2x Factoring out the y on the left side: y(8x^2+3) = 2x Dividing both sides by (8x^2+3): y = 2x/(8x^2+3)
 expressions/210905: This question is from textbook Practical Business Math 4(P-9)=64 The answer in the back states P=25 I can't figure out how they got this? Thanks for your help1 solutions Answer 159315 by nerdybill(6958)   on 2009-09-03 23:22:58 (Show Source): You can put this solution on YOUR website!4(P-9)=64 Strategy, isolate P: . Begin by dividing both sides by 4: (P-9)=16 P-9 = 16 Add 9 to both sides: P = 25 .
 Rational-functions/210900: sorry i got it off a worksheet: f(x)=x^2 + 2x + 5 i have to write each quadratic function in vertex form, give coordinates of the vertex, and the equation of the axis of symmetry.1 solutions Answer 159314 by nerdybill(6958)   on 2009-09-03 23:20:16 (Show Source): You can put this solution on YOUR website!f(x)=x^2 + 2x + 5 . The axis of symmetry is the line x = -b/2a x = -2/2 = -1 x = -1 (equation of axis of symmetry) . f(x)=x^2 + 2x + 5 f(x)=(x^2 + 2x) + 5 f(x)=(x^2 + 2x + __ ) + 5 f(x)=(x^2 + 2x + 1 ) + 5 - 1 f(x)=(x+1)^2 + 4 f(x)=(x-(-1))^2 + 4 (vertex form) . vertex is at (-1,4)
 Percentage-and-ratio-word-problems/210898: A woman earns 15% more than her husband. Together they make \$69,875 per year. What is the husband's annual salary?1 solutions Answer 159313 by nerdybill(6958)   on 2009-09-03 23:12:31 (Show Source): You can put this solution on YOUR website! A woman earns 15% more than her husband. Together they make \$69,875 per year. What is the husband's annual salary? . Let x = husband's annual salary then 1.15x = wife's annual salary . x + 1.15x = 69875 2.15x = 69875 x = 69875/2.15 x = \$32,500 (husband's salary)