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# Recent problems solved by 'nerdybill'

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 Miscellaneous_Word_Problems/373121: I am not sure how to go about this word problem so can you please help me solve this problem. I am getting overwhelmed! John wants to fence a 150 square meters rectangular field. He wants the length and width to be natural numbers {1,2,3,...}. What field dimensions will require the least amount of fencing? Show work. Thank you so much! Jessica1 solutions Answer 265651 by nerdybill(6963)   on 2010-11-18 10:32:10 (Show Source): You can put this solution on YOUR website!John wants to fence a 150 square meters rectangular field. He wants the length and width to be natural numbers {1,2,3,...}. What field dimensions will require the least amount of fencing? . You could create a "table": width length area perimeter 1 150 150 302 2 75 150 154 3 50 150 106 5 30 150 70 6 25 150 62 10 15 150 50 15 10 150 50 25 6 150 62 30 5 150 70 . From the above, we see that the answer is a field that is: 10 feet by 15 feet
 Exponential-and-logarithmic-functions/373122: I don't even know where to start. The problem is log2^x = 1/3log2^27. Can you please show me step by step.1 solutions Answer 265648 by nerdybill(6963)   on 2010-11-18 10:14:56 (Show Source): You can put this solution on YOUR website!Apply "log rules"... . Starting with:
 Equations/373108: Hi tutors.. :) Im having a problem with Algebra Number Problems. For example an equation like this : There are two numbers whose sum is 72. One number is twice the other. What are the numbers? Also. There are three consecutive integers. the sum of the smallest and largest is 36. I just dont get it:( Help please1 solutions Answer 265647 by nerdybill(6963)   on 2010-11-18 10:11:15 (Show Source): You can put this solution on YOUR website!There are two numbers whose sum is 72. One number is twice the other. What are the numbers? . Let x = first number then from "One number is twice the other." we have 2x = second number . From: "There are two numbers whose sum is 72" we derive our equation: x + 2x = 72 3x = 72 x = 24 (first number) Second number: 2x = 2(24) = 48 . . . Also. There are three consecutive integers. the sum of the smallest and largest is 36. . Let x = first consecutive integer then x+1 = second consecutive integer x+2 = third . From:"the sum of the smallest and largest is 36" we get: x + x+2 = 36 2x+2 = 36 2x = 34 x = 17 (first) . Second: x+1 = 17+1 = 18 . Third: x+2 = 17+2 = 19
 Quadratic_Equations/372797: Jack wanted to throw an apple to Lauren, who was on a balcony 40 feet above him, so he tossed it upward with an initial speed of 56 ft/s. Lauren missed it on the way up, but then caught it on the way down. How long was the apple in the air? Would I use the equation h(t) = -16t^2 + 56t +40? (where h = height and t = time) I'm not very good at solving quadratic word problems :(1 solutions Answer 265524 by nerdybill(6963)   on 2010-11-17 18:32:32 (Show Source): You can put this solution on YOUR website!Well, you're close The general equation is h(t) = -(1/2)gt^2 + Vot + h(0) where g is gravity 32 ft/sec^2 Vo is initial velocity h(0) is initial height of Jack (not Lauren) . So, plugging in what we know h(t) = -(1/2)(32)t^2 + 56t + 0 which reduces to h(t) = -16t^2 + 56t . So, since Lauren is 40 feet up -- set h(t) to 40 and solve for t: h(t) = -16t^2 + 56t 40 = -16t^2 + 56t 0 = -16t^2 + 56t - 40 multiply both sides by -1: 0 = 16t^2 - 56t + 40 divide both sides by 8: 0 = 2t^2 - 7t + 5 0 = (2t-5)(t-1) So, t = {1, 5/2} 1 sec represents the time the apple reaches 40 ft (going up) So 5/2 secs or 2 and 1/2 secs (is your solution)
 logarithm/372785: 3logx- log11- 4logz+2logy in condensed form1 solutions Answer 265516 by nerdybill(6963)   on 2010-11-17 18:23:41 (Show Source): You can put this solution on YOUR website!3logx- log11- 4logz+2logy logx^3 - log11- logz^4 + logy^2 log(x^3/11) - logz^4 + logy^2 log(x^3/11)/z^4 + logy^2 log(x^3/11z^4) + logy^2 log(x^3/11z^4)y^2 log(x^3y^2/11z^4)
 Linear-systems/372684: If an object is projected vertically upward from an altitude of s(o) feet with an intial velocity of v(o) ft/sec, then its distance above the ground after t seconds is: s(t)=-16t^2+ v(o)t+ s(o). If s(1)= 64 and s(2)=66, what are s(o) and v(o)? I honestly do not know how to set the problem up! Please help, it would be greatly appreciated!Thanks :) 1 solutions Answer 265418 by nerdybill(6963)   on 2010-11-17 14:26:23 (Show Source): You can put this solution on YOUR website!Given: s(t)=-16t^2+ v(o)t+ s(o). If s(1)= 64 and s(2)=66, what are s(o) and v(o)? . So, when x=1 then s(1)=64 and when x=2 then s(2)=66 . From here, you now have two unknowns and two equations: when x=1 then s(1)=64 s(t)=-16t^2+ v(o)t+ s(o) s(1)=-16(1)^2+ v(o)(1)+ s(o) 64=-16+ v(o)+ s(o) 89=v(o)+ s(o) equation 1 . when x=2 then s(2)=66 s(t)=-16t^2+ v(o)t+ s(o) s(2)=-16(2)^2+ v(o)(2)+ s(o) 66=-16(4)+ 2v(o)+ s(o) 130=2v(o)+ s(o) equation 2 . Considering the "system of equations" then: 130=2v(o)+ s(o) 89=v(o)+ s(o) . Multiply both sides of the bottom equation by -1 and add: 130=2v(o)+ s(o) -89=-v(o)- s(o) --------------------- 41 ft/sec = v(0) . Substitute the above into: 89=v(o)+ s(o) 89=41+ s(o) 48 feet = s(o)
 Polynomials-and-rational-expressions/372538: I am having a problem rearranging polynomials so that I can find the GCF and factor by grouping to get a binomial answer. Example question: 10x^2-12y+15x-8xy. I have tried to rearrange by switching the 12y and 15x but it doesn't give me the correct binomial answer. Can you advise? I hope I haven't confused anyone with my question. I appreciate any help. Juliette Wilson1 solutions Answer 265332 by nerdybill(6963)   on 2010-11-17 09:27:42 (Show Source): You can put this solution on YOUR website!Starting with: 10x^2-12y+15x-8xy Then switching the two middle terms: 10x^2+15x-12y-8xy Now group terms: (10x^2+15x)-(12y+8xy) (careful with the sign change in second term) factor each expression inside parentheses: 5x(2x+3)-4y(3+2x) 5x(2x+3)-4y(2x+3) (2x+3)(5x-4y)
 logarithm/371941: Solve the following logarithmic equation. Check all solutions. log(x)+log(x+21)=21 solutions Answer 264953 by nerdybill(6963)   on 2010-11-16 10:55:57 (Show Source): You can put this solution on YOUR website!log(x)+log(x+21)=2 log(x(x+21))=2 x(x+21) = 10^2 x^2+21x = 100 x^2+21x-100 = 0 Factoring the left: (x+25)(x-4) = 0 . x = {-25, 4} We can throw out the -25 solution -- it is an extraneous solution leaving: x = 4
 Quadratic_Equations/371787: Find the minimum value of g(x)=5x^2+17x-31 solutions Answer 264818 by nerdybill(6963)   on 2010-11-15 23:25:50 (Show Source): You can put this solution on YOUR website!g(x)=5x^2+17x-3 . Because it is in the form of Ax^2 + Bx + C it is a "quadratic" -- a parabola By looking at the coefficient associated with the x^2 term we can tell whether it opens upwards (positive coefficient) or downwards (negative coefficient). It is positive so it opens upwards. . Finding the vertex give you the minimum. it is minimum when x = -b/(2a) = -17/(2*5) = -17/10 = -1.7 . Plug it back into the original equation to find the minimum: g(x)=5x^2+17x-3 g(-1.7)=5(-1.7)^2+17(-1.7)-3 g(-1.7)= -17.45
 Polynomials-and-rational-expressions/371282: Factor: 6x^3+6x^2+x+11 solutions Answer 264504 by nerdybill(6963)   on 2010-11-15 09:41:18 (Show Source): You can put this solution on YOUR website!6x^3+6x^2+x+1 Group terms: (6x^3+6x^2)+(x+1) Factoring left term: 6x^2(x+1)+(x+1) Factoring out (x+1): (x+1)(6x^2 + 1)
Rate-of-work-word-problems/371009: Aubrey uses a slingshot to shoot a marble from a window. The height of the marble h(t) in feet at time t seconds is given by the function h(t)=-16t square -50t+164. How many seconds elapse before the marble hits the ground?
1 solutions

Answer 264335 by nerdybill(6963)   on 2010-11-14 19:23:17 (Show Source):
You can put this solution on YOUR website!
h(t)=-16t square -50t+164
When it reaches the ground the "height" is zero
so, set h(t) to zero and solve for t:
0=-16t^2 -50t+164
0=16t^2 + 50t - 164
0=8t^2 + 25t - 82
Applying the "quadratic formula" we find that
t = {2, -5.125}
Throw out the negative answer leaving
t = 2 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=3249 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 2, -5.125. Here's your graph:

 Equations/370764: Here is a word problem didn't know if i set it up right and solved it right Three is added to a number and then the sum is multiplied by four, the result is 16 I got (n+3)4=16 n=3.25 Is this correct because it doesn't work out when you replace it with n 1 solutions Answer 264273 by nerdybill(6963)   on 2010-11-14 12:29:28 (Show Source): You can put this solution on YOUR website!Your equation is correct: (n+3)4=16 . To solve for n, we start by dividing both sides by 4: [(n+3)4]/4 = 16/4 (n+3) = 4 n+3 = 4 subtracting 3 from both sides: n = 1
 Linear-equations/370482: What is the slope of a line that contains the point (2,-3) and has the same x-intercept as x+5y=8?1 solutions Answer 264094 by nerdybill(6963)   on 2010-11-13 18:28:02 (Show Source): You can put this solution on YOUR website!What is the slope of a line that contains the point (2,-3) and has the same x-intercept as x+5y=8? . First, find x-intercept of x+5y=8 Do this by setting y=0 and solve for y: x+5y=8 0+5y=8 y = 8/5 So, the x-intercept is at (0, 8/5) . Now, we determine the slope between two points: (2,-3) and (0, 8/5) . slope = (y2-y1)/(x2-x1) slope = (8/5-(-3))/(0-2) slope = (8/5+3)/(-2) slope = (8/5+15/5)/(-2) slope = (23/5)/(-2) slope = -23/(5*2) slope = -23/10 slope = -2.3
 logarithm/369739: how to you write 3 times the natural logarithm of x to the third times y plus 2 times the natural logarithm of y times z to the third as a single natural logarithm? so 3*ln(x^(3)y) + 2*ln(yz^2) I know using the power rule that the 3 and 2 can be moved to ln(x^(3)y)^3 + ln(yz^2)^2 i also know about the product rule but i don't know how far this can be simplified because of the parentheses around the (x^(3)y) and (yz^2). 1 solutions Answer 263493 by nerdybill(6963)   on 2010-11-11 17:44:54 (Show Source): You can put this solution on YOUR website!so 3*ln(x^(3)y) + 2*ln(yz^2) I know using the power rule that the 3 and 2 can be moved to ln(x^(3)y)^3 + ln(yz^2)^2 . So far, so good! Next, we distribute the exponent to terms inside the parentheses: ln((x^3)^3y^3) + ln(y^2(z^2)^2) ln(x^9y^3) + ln(y^2z^4) ln((x^9y^3)(y^2z^4)) ln(x^9y^5z^4)
 Exponential-and-logarithmic-functions/369722: Solve for x. log x-log 3=21 solutions Answer 263484 by nerdybill(6963)   on 2010-11-11 17:20:58 (Show Source): You can put this solution on YOUR website!log x-log 3=2 log x/3=2 x/3 = 10^2 x/3 = 100 x = 300
 Exponential-and-logarithmic-functions/369724: Solve for x. e^-3x+2=401 solutions Answer 263483 by nerdybill(6963)   on 2010-11-11 17:19:44 (Show Source): You can put this solution on YOUR website!Not sure whether you meant: Or . I'll assume the later: take the ln of both sides to give:
 test/369648: While pipe A can fill a tank in 3 hours, pipe B can take out a full tank in fours. A full tank fits 360 gallons of water. If both pipes are turned on at the same time, how many gallons of water would be in the tank in 6 hours?1 solutions Answer 263481 by nerdybill(6963)   on 2010-11-11 16:57:23 (Show Source): You can put this solution on YOUR website!While pipe A can fill a tank in 3 hours, pipe B can take out a full tank in fours. A full tank fits 360 gallons of water. If both pipes are turned on at the same time, how many gallons of water would be in the tank in 6 hours? . Rate of pipe A: 360 gallons per 3 hours Rate of pipe B: 360 gallons per 4 hours . Gallons in 6 hours: 6(360/3 + 360/4) 6(120 + 90) 6(220) 13320 gallons . This appears to be a trick question. Since the tank only holds 360 gallons, if both pipe A and B were turned on for 6 hours, 13320 gallons would come from the pipe, 960 gallons would overflow the tank and only 360 gallons would remain in the tank.
 Reduction-of-unit-multipliers/369681: Use three unit multipliers to convert 36 yards to centimeters1 solutions Answer 263436 by nerdybill(6963)   on 2010-11-11 15:43:16 (Show Source): You can put this solution on YOUR website! Use three unit multipliers to convert 36 yards to centimeters . 36 yards * 3 feet/yard * 12 inches/feet * 2.54 cm/inch = 36*3*12*2.54 = 3291.84 centimeters
 Circles/369568: If a circle with a radius of 10 has its radius decreased by 2 , by what percent is the area decreased? A. 20% B. 25%C. 36% D. 40% E. none1 solutions Answer 263354 by nerdybill(6963)   on 2010-11-11 12:18:20 (Show Source): You can put this solution on YOUR website!If a circle with a radius of 10 has its radius decreased by 2 , by what percent is the area decreased? A. 20% B. 25%C. 36% D. 40% E. none . Since area of any circle: (pi)r^2 . 10 inch radius circle: (pi)10^2 = 100(pi) . 8 inch radius circle: (pi)8^2 = 64(pi) . Decrease: 100(pi) - 64(pi) = 36(pi) . Percent of decrease: [36(pi)]/[100(pi)] * 100 36/100 * 100 36%
 Miscellaneous_Word_Problems/369490: Ashley is making punch for a party. The recipe requires twice as much orange juice as cranberry juice and 8 times as much ginger ale as cranberry juice. If she plans to make 176 onces of punch, how much of each ingredient should she use.1 solutions Answer 263313 by nerdybill(6963)   on 2010-11-11 09:56:42 (Show Source): You can put this solution on YOUR website!Ashley is making punch for a party. The recipe requires twice as much orange juice as cranberry juice and 8 times as much ginger ale as cranberry juice. If she plans to make 176 onces of punch, how much of each ingredient should she use. . Let x = onces of cranberry juice then 2x = onces of orange juice 8x = onces of ginger ale . x + 2x + 8x = 176 11x = 176 x = 176/11 x = 16 onces (cranberry juice) . Orange juice: 2x = 2(16) = 32 ounces . Ginger ale: 8x = 8(16) = 128 ounces
 logarithm/369027: I have 16^(3/2)=64 and i have to put it in log form. I got the answer as log(base16)64=3/2. But it does not check, what am i doing wrong?1 solutions Answer 262987 by nerdybill(6963)   on 2010-11-10 11:18:08 (Show Source): You can put this solution on YOUR website!If you started with: taking log base 16 of both sides: cross multiplying ... easier way would be to:
 Equations/368989: A tennis ball is rolled down a valley along a slope shaped like a parabola. After t seconds, its height above sea level in meters is modeled by the function h(t)=t^2-8.9t+14. After how many seconds does the ball reach its minimum height? What is that minimum height? Round to the nearest hundredths place and use proper units in your answer.1 solutions Answer 262961 by nerdybill(6963)   on 2010-11-10 09:58:28 (Show Source): You can put this solution on YOUR website!h(t)=t^2-8.9t+14 . We know it is a "quadratic" because it is in the form of: Ax^2 + Bx + C . Looking at the coefficient associated with the t^2 term, since it is positive (happy face) the parabola opens upwards. The vertex will give you the minimum. . After how many seconds does the ball reach its minimum height? t = -b/(2a) t = -(-8.9)/(2*1) t = (8.9)/2 t = 4.45 secs . What is that minimum height? Plug the value above back into: h(t)=t^2-8.9t+14 h(4.45)=4.45^2-8.9(4.45)+14 h(4.45)= -5.80 meters
 Rate-of-work-word-problems/368342: Working alone, Ryan can dig a 10 ft by 10 ft hole in five hours. Castel can dig the same hole in six hours. How long would it take them if they worked together?1 solutions Answer 262492 by nerdybill(6963)   on 2010-11-09 09:30:01 (Show Source): You can put this solution on YOUR website!Working alone, Ryan can dig a 10 ft by 10 ft hole in five hours. Castel can dig the same hole in six hours. How long would it take them if they worked together? . Let x = time (hours) it takes for them to dig together then x(1/5 + 1/6) = 1 Multiplying both sides by 30: x(6 + 5) = 30 11x = 30 x = 30/11 x = 2.73 hours or x = 2 hours and 44 minutes
 logarithm/368021: solve the equation: 2 ^logx = 81 solutions Answer 262275 by nerdybill(6963)   on 2010-11-08 18:18:12 (Show Source): You can put this solution on YOUR website!2^logx = 8 2^logx = 2^3 take log base 2 of both sides: logx = 3 x = 10^3 x = 1000
 logarithm/368020: solve the equation: log4 + 2logx =log (x+3)^21 solutions Answer 262274 by nerdybill(6963)   on 2010-11-08 18:16:24 (Show Source): You can put this solution on YOUR website!log4 + 2logx =log (x+3)^2 log4 + logx^2 =log (x+3)^2 log(4x^2) =log (x+3)^2 4x^2 =(x+3)^2 2x =x+3 x = 3
 logarithm/368024: solve for x : log8+3logx=log2x1 solutions Answer 262273 by nerdybill(6963)   on 2010-11-08 18:14:46 (Show Source): You can put this solution on YOUR website!I'm assuming everything is log base 10... log8+3logx=log2x log8+logx^3=log2x log(8x^3)=log2x 8x^3 = 2x 8x^2 = 2 x^2 = 2/8 x^2 = 1/4 x = 1/2
 Numbers_Word_Problems/368038: The sum of two numbers is 48 and their difference is 80. Find each ofthe number.1 solutions Answer 262272 by nerdybill(6963)   on 2010-11-08 18:12:22 (Show Source): You can put this solution on YOUR website!The sum of two numbers is 48 and their difference is 80. Find each ofthe number. Let x = one number and y = the other number . Our "system" of equations: x+y = 48 x-y = 80 . So, applying the "addition method" we add the two equations together: x+y = 48 x-y = 80 --------- 2x = 128 x = 64 (one number) . find other number by substituting above into first equation: x+y = 48 64+y = 48 y = 48-64 y = -16 (other number)
 Triangles/367853: I have a ramp or slope at 65 feet long. It rises up at 32 degrees. what is the height of the ramp from earth at the end . ?? Thank you.1 solutions Answer 262190 by nerdybill(6963)   on 2010-11-08 14:59:54 (Show Source): You can put this solution on YOUR website!I have a ramp or slope at 65 feet long. It rises up at 32 degrees. what is the height of the ramp from earth at the end . ?? . Applying a little trig... Let h = height of ramp at end then since tan = opposite/hypotenuse tan(32)= h/65 65*tan(32)= h 40.62 feet = h
 Rate-of-work-word-problems/367854: Timothy can mow his lawn in 40 minutes, but his brother Shamus needs only 30 minutes to mow the same lawn. If they work together on the lawn, how long will it take them to finish mowing?1 solutions Answer 262189 by nerdybill(6963)   on 2010-11-08 14:56:38 (Show Source): You can put this solution on YOUR website!Timothy can mow his lawn in 40 minutes, but his brother Shamus needs only 30 minutes to mow the same lawn. If they work together on the lawn, how long will it take them to finish mowing? . Let x = time (minutes) for both to mow the lawn then x(1/40 + 1/30) = 1 Multiplying both sides by (40)(30): x(30 + 40) = 1200 70x = 1200 x = 1200/70 x = 17.143 minutes Or, x = 17 minutes (approximately)
 Exponents/367505: Graph the quadratic function ; state the ordered pairs representing the x-intercept(s), y-intercept (if any), and the vertex of the parabola.1 solutions Answer 261963 by nerdybill(6963)   on 2010-11-07 22:35:37 (Show Source): You can put this solution on YOUR website! To find the y-intercept set x=0 and solve for f(x): y-intercept is at (0, 15) . To find the x-intercept set f(x)=0 and solve for x: x = {-3, 5} x-intercepts are at (-3,0) and (5,0) . Vertex: x-coordinate: x = -b/(2a) = -(2)/(2*(-1)) = -2/(-2) = 1 y-coordinate: vertex is at (1, 16)
 Graphs/367297: Perpendicular to the graph of y = 1/3 x + 7; y-intercept (0,5)1 solutions Answer 261818 by nerdybill(6963)   on 2010-11-07 16:19:41 (Show Source): You can put this solution on YOUR website! Perpendicular to the graph of y = 1/3 x + 7; y-intercept (0,5) .56 Because y = 1/3 x + 7 is in "slope-intercept" form: y=mx+b We can see that the slope is: 1/3 . For a line to be perpendicular, the slope of the new line must be the "negative reciprocal": (1/3)m = -1 m = -3 (slope of our new line) . Given that the problem stated that the y-intercept is at (0,5) this means that the 'b' (in the slope intercept form) is 5 . So, our equation is: y = -3x + 5 (this is what they're looking for)