Algebra ->  Tutoring on algebra.com -> See tutors' answers!      Log On

 Tutoring Home For Students Tools for Tutors Our Tutors Register Recently Solved
 By Tutor
| By Problem Number |

Tutor:

# Recent problems solved by 'Theo'

Jump to solutions: 0..29 , 30..59 , 60..89 , 90..119 , 120..149 , 150..179 , 180..209 , 210..239 , 240..269 , 270..299 , 300..329 , 330..359 , 360..389 , 390..419 , 420..449 , 450..479 , 480..509 , 510..539 , 540..569 , 570..599 , 600..629 , 630..659 , 660..689 , 690..719 , 720..749 , 750..779 , 780..809 , 810..839 , 840..869 , 870..899 , 900..929 , 930..959 , 960..989 , 990..1019 , 1020..1049 , 1050..1079 , 1080..1109 , 1110..1139 , 1140..1169 , 1170..1199 , 1200..1229 , 1230..1259 , 1260..1289 , 1290..1319 , 1320..1349 , 1350..1379 , 1380..1409 , 1410..1439 , 1440..1469 , 1470..1499 , 1500..1529 , 1530..1559 , 1560..1589 , 1590..1619 , 1620..1649 , 1650..1679 , 1680..1709 , 1710..1739 , 1740..1769 , 1770..1799 , 1800..1829 , 1830..1859 , 1860..1889 , 1890..1919 , 1920..1949 , 1950..1979 , 1980..2009 , 2010..2039 , 2040..2069 , 2070..2099 , 2100..2129 , 2130..2159 , 2160..2189 , 2190..2219 , 2220..2249 , 2250..2279 , 2280..2309 , 2310..2339 , 2340..2369 , 2370..2399 , 2400..2429 , 2430..2459 , 2460..2489 , 2490..2519 , 2520..2549 , 2550..2579 , 2580..2609 , 2610..2639 , 2640..2669 , 2670..2699 , 2700..2729 , 2730..2759 , 2760..2789 , 2790..2819 , 2820..2849 , 2850..2879 , 2880..2909 , 2910..2939 , 2940..2969 , 2970..2999 , 3000..3029 , 3030..3059 , 3060..3089 , 3090..3119 , 3120..3149 , 3150..3179 , 3180..3209 , 3210..3239 , 3240..3269 , 3270..3299 , 3300..3329 , 3330..3359 , 3360..3389 , 3390..3419 , 3420..3449 , 3450..3479, >>Next

 Triangles/251486: In a triangle, the lengths of three medians are 4, 5 and 6, find the area of the triangle.1 solutions Answer 183155 by Theo(3458)   on 2009-12-19 08:59:09 (Show Source): You can put this solution on YOUR website!area of a triangle in terms of the median is expressed as follows: The area of a triangle can be expressed in terms of the medians by: where: . see http://mathworld.wolfram.com/TriangleMedian.html for details. in terms of your problem: m1 = 4 m2 = 5 m3 = 6 area of the triangle is therefore: this becomes: which becomes: which becomes: which becomes A = 13.22875656 here's another reference: http://www.math10.com/en/geometry/median.html here's yet another. http://pballew.net/medians.htm
 percentage/251300: A man gets a rise of 10% in salary at the end of his first year of service, and further rise of 20% and 25 % at the end of the second and third years respectively, the rise in each case being calculated on his salary at the beginning of the year. To what annual percentage increase is this equivalent?1 solutions Answer 183148 by Theo(3458)   on 2009-12-19 06:54:37 (Show Source): You can put this solution on YOUR website!assume his salary starts at 1. at the beginning of the first year he is making 1. at the end of the first year he is making 1 * 1.1 = 1.1 at the end of the second year he is making 1.1 * 1.2 = 1.32 at the end of the third year he is making 1.32 * 1.25 = 1.65 he starts at 1 and at the end of the third year he is making 1.65 his average annual increase is given by the compounding formula of: 1.65 = 1*(1+x)^3 this is equivalent to: 1.65 = (1+x)^3 take the cube root of both sides to get: (1.65)^(1/3) = 1+x subtract 1 from both sides to get: x = (1.65)^(1/3) - 1 solve for x to get: x = .18166575 at the beginning of the first year he is making 1. at the end of the first year he is making 1 * 1.18166575 = 1.18166575 at the end of the second year he is making 1.18166575 * 1.18166575 = 1.396333946 at the endof the third year he is making 1.396333946 * 1.18166575 = 165 1 * (1.18166575)^3 = 1.65 the equation that was used was the future value of a present amount formula that is equal to: f = p * (1+i)^n f = future value p = present amount i = interest rate per time period n = number of time periods. to solve this problem, we first had to find f. that was done using the year by year analysis up top. once we knew f, we could then substitute in the formula to get: f = 1.65 p = 1 i = x n = 3 we then solved for x.
 Mixture_Word_Problems/251464: Mixture A is 7.5% acid. To raise the concentration of acid to 10%, some pure acid will be added to the mixture. How many liters of mixture A and how many liters of pure acid are needed to end up with 200 liters of 10% solution. Not sure if I started it out right, that's why I'm here! x = pure acid A = Mixture A 7.5 + x = (200)(.10) x = 12.5 liters If that's right, I'm not sure where to go from there. If you can help me, that would be great! Thank you.1 solutions Answer 183147 by Theo(3458)   on 2009-12-19 06:44:24 (Show Source): You can put this solution on YOUR website!mixture A is 7.5% acid. let x = amount of mixture A. 7.5% of x is equivalent to .075 * x (you have to divide percent by 100% to get proportion). this means that .075 * x = amount of acid in mixture A. let y = amount of mixture C. then 1.0 * y = amount of acid in mixture B. this is because 100% of mixture B is equivalent to 1.0 * mixture B. mixture C is equal to 200 liters. this means that: x + y = 200 mixture C will contain 10% acid which means that mixture C will contain .10 * 200 = 20 liters of acid. this means that: .075 * x + y = 20 you have 2 equations that need to be solved simultaneously. they are: x + y = 200 (first equation) .075*x + y = 20 (second equation) you can solve for y in either equation and then solve for x in the other equation. we'll solve for y in the first equation to get: y = 200-x we'll substitute for y in the second equation to get: .075*x + (200-x) = 20 remove parentheses to get: .075*x + 200 - x = 20 combine like terms to get: -.925*x + 200 = 20 subtract 200 from both sides to get: -.925*x = 20-100 = -180 divide both sides by -.925 to get: x = -180 / -.925 = 194.5945946 use this value of x to solve for y in the first equation to get: y = 200 - 194.5945946 = 5.405405405 use the values for x and y in the second equation to confirm that they are good. the second equation is: .075*x + y = 20 substituting for x and y in that equation, we get: .075*194.5945946 + 5.405405405 = 20 this becomes 20 = 20 which is true confirming our values for x and y are good. your answer is: you need to add 5.405405405 liters of pure acid to make a 200 liter mixture of 10% acid.
 Miscellaneous_Word_Problems/251474: We just blew some air into a spherical balloon and doubled its volume. By how much did we multiply the surface area ?1 solutions Answer 183144 by Theo(3458)   on 2009-12-19 06:10:56 (Show Source): You can put this solution on YOUR website!formula for volume of a sphere is: formula for surface area of a sphere is: if we solve for r in the equation for the volume of the sphere, we get: take the cube root of both sides of this equation to get: if substitute for the value of r in the equation for the surface area of the sphere, then we get: becomes: if we double v, then this equation becomes: if we divide s[2] by s[1] we get: this simplifies to: which is the same as: multiply both sides of this equation by and you get: for example: let the volume of the sphere be equal to 300. we solve for the radius to get: radius of the sphere equals 4.152830592 we confirm by substituting in the equation for the volume of the sphere to get: v = 4/3 * pi * (4.152830592)^3 = 300 cubic units. the surface area of the sphere is equal to 4 * pi * r^2 = 216.7196518 square units. if the sphere doubles in volume, then the surface area of the new sphere should be equal to 216.7196518 * making the surface area of the new sphere equal to square units to confirm this is correct, we need to solve for the radius of the new sphere. the volume of the new sphere is equal to 600 cubic units. the radius of the new sphere is calculated to be 5.23223868 units. the surface area of the new sphere is equal to 4 * pi * r^2 which becomes since this is the same as then we're good. the answer to your question is: the surface area was multiplied by .
 Miscellaneous_Word_Problems/251476: If the diameter of a cylindrical can is increased by 30 percent, by approximating what percentage should the height be increased to triple the volume of the can ?1 solutions Answer 183143 by Theo(3458)   on 2009-12-19 04:41:45 (Show Source): You can put this solution on YOUR website!if the diameter of the can is increased by 30%, then the radius of the can is also increased by 30%. consider. d2 = d1 * 1.3 = 1.3*d1 d1 = r1*2 d2 = 1.3*d1 = 1.3*2*r1 r2 = d2/2 = 1.3*2*r1/2 = 1.3*r1 volume of the can is equal to pi*r^2*h increase the radius by 30% and you get volume of the can is equal to pi*(1.3*r)^2*h. in order for the volume of the enlarged radius can to equal 3 times the volume of the original can, the following equation needs to be satisfied. let x = the amount the height of the can has to be increased. then: let v[1] = the volume of the original can. let v[2] = the volume of the can with a 30% increase in the length of the radius and an unknown increase in the height of the can. let x = the unknown increase in the height of the can. v[1] = pi*r^2*h v[2] = pi*(1.3*r)^2*x*h = 3 * v[1] = 3 * pi*r^2*h this results in: pi * (1.3*r)^2 * x * h = 3 * pi * r^2 * h you want to solve for x. divide both sides of this equation by h to get: pi * (1.3*r)^2 * x = 3 * pi * r^2 divide both sides of this equation by (pi * (1.3*r)^2 to get: x = 3 * pi * r^2 / (pi * (1.3*r)^2) (1.3*r)^2 becomes (1.3)^2 * r^2. substitute in equation to get: x = 3 * pi * r^2 / (pi * (1.3)^2 * r^2) pi and r^2 cancel out from numerator and denominator to get: x = 3 / (1.3)^2 = 1.775147929 to make the equations equivalent, the height needs to be multiplied by 1.775147929 which is the same as (3/(1.3)^2). substitute in the original equation to confirm that this is true. the original equation is: pi*(1.3*r)^2*x*h = 3 * pi*r^2*h replace x with (3/(1.3)^2) to get: pi*(1.3*r)^2*(3/(1.3)^2)*h = 3 * pi*r^2*h simplify by removing parentheses to get: pi * (1.3)^2 * r^2 * 3 * h / (1.3)^2 = 3 * pi * r^2 * h (1.3)^2 in numerator and denominator cancel out to get: pi * r^2 * 3 * h = 3 * pi * r^2 * h except for the order in which the terms are presented, the expressions on each side of the equal side are identical confirming that the value of x = (3/(1.3)^2) is correct. your answer is: the height needs to be multiplied by (3/(1.3)^2) = 1.775147929 in order for the volume of the resulting can to be tripled, assuming that the diameter has been increased by 30%. example: let r = 15 let h = 22 volume of the original can is pi*r^2*h = pi*(15)^2*22 = 1550.88364 multiply radius by 1.3 to get r = 19.5 multiply height by (3/(1.3)^2) to get h = 39.05325444 volume of the enlarged can is pi*r^2*h = pi*(19.5)^2*39.05325444 = 46652.65091. 46652.65091 / 1550.88364 = 3 confirming the value of x is good. the answer to the question is that the height should be increased by approximately 77.51%. the original height is x. the new height is 1.775147929 times x new height minus old height times 100% equals the percent increase. 1.775147929 * x - x = .775147929 * x * 100% equals 77.5147929% * x = ~ 77.51% * x. ~ means approximately after rounding to the nearest hundredth of a percent. the height needs to be increased by approximately 77.51%.
 expressions/251304: please help me solve this equation: (-6/y)^3 1 solutions Answer 182980 by Theo(3458)   on 2009-12-18 09:00:42 (Show Source): You can put this solution on YOUR website!(-6/y)^3 = (-6)^3/y^3 = -126/y^3 this is because (a/b)^3 = a^3/b^3 let a = 5 and let b = 9 (a/b)^3 = (5/9)^3 = .171467764 (use your calculator to solve) (a/b)^3 = a^3/b^3 = 5^3/9^3 = 125/729 = .171467764 they are equivalent. what you were asking me to solve is not an equation, but an expression. (-6/y)^3 is an expression. when you make it equal to something, then the whole thing becomes an equation. example: (-6/y)^3 = 99635 now you have an equation. the expression on the left side of the equal sign equals the expression on the right side of the equal sign.
 Polynomials-and-rational-expressions/251293: Divide and if possible, Simplify. (a+7)divided by 3a^2+14a-49(over)a^2+8a+7 This was my attempt but teacher said it was wrong. Can you please help me so I can see where I went wrong? (a+7)/ 3a^2+14a-49 over a^2+8a+7 (a+7)/[(3a-7)(a+7)] over [(a+7)(a+1)] Cancel(a+7) to get 1/[(3a-7)] over [(a+7)(a+1)] Rewrite [1/(3a-7)]/[(a+7)(a+1)]/1 Invert the denominator and multiply = 1/[(3a-7)(a+7)(a+1)] 1 solutions Answer 182978 by Theo(3458)   on 2009-12-18 08:54:26 (Show Source): You can put this solution on YOUR website!looks like you got the factors ok. your original equation is: you factored correctly to get: you canceled correctly to get: it looks like where you went wrong was when you inverted the denominator. since = , your answer should have been: which becomes:
 Percentage-and-ratio-word-problems/250952: Before conference swim championships, Suzanne’s best time in the 200 yard butterfly race was 2 minutes, 23 seconds. In the conference championships, she won 10th place by swimming the 200 yard butterfly in 2 minutes, 15 seconds. What per cent decrease is that? A.3.6% B.5.6% C.5.9% D.6.2% E.36.9% 1 solutions Answer 182730 by Theo(3458)   on 2009-12-17 08:27:36 (Show Source): You can put this solution on YOUR website!2 minutes 23 seconds is equivalent to 143 seconds. 2 minutes 15 seconds is equivalent to 135 seconds. she lowered her time by 8 seconds (135 + 8 = 143). the percent decrease equals the difference divided by the first time, with the result of that operation being multiplied by 100%. This becomes: 8/143 = .05594405594 * 100% = 5.594405594 percent. that's pretty close to selection B of 5.6%
 Geometry_Word_Problems/250878: A rectangular fish tank is 2 feet high and has a volume of 48 feet cubed. What could be a possible set of whole number dimensions for the base of the tank?1 solutions Answer 182724 by Theo(3458)   on 2009-12-17 08:04:47 (Show Source): You can put this solution on YOUR website!volume = height * area of base. let v = volume, h = height, a = area of base. v = h*a solve for a to get: a = v/h v = 48 h = 2 a = 48/2 = 24 the area of the base is 24 square inches. 24 = 1*24 = 2*12 = 3*8 = 4*6 possible dimensions for the base where the dimensions are in whole numbers are: ``` length width 24 1 12 2 8 3 6 4 ```
 Exponential-and-logarithmic-functions/250880: A piece of charcoal is found to contain 30% of the carbon 14 that it originally had. When did the tree die from which the charcoal came? Use 5600 years as the half-life of carbon 14.1 solutions Answer 182721 by Theo(3458)   on 2009-12-17 07:53:47 (Show Source): You can put this solution on YOUR website!formula is: F = P*e^(kt) where: F = future amount P = present amount k = constant of proportion. t = time half life of carbon is given by the equation: .5 = e^(5600k) because: half life of carbon is 5600 years. solve for k and then you can solve the problem. take log of both sides of equation of .5 = e^(5600k) to get: log(.5) = log(e^(5600k)) this becomes: log(.5) = 5600k * log(e) divide both sides of this equation by 5600*log(e) to get: k = log(.5)/(5600*log(e)) solve for k to get: k = -.301029996 / (5600*(.434294482) which becomes: k = -.000123776 substitute in your original equation to confirm this value for k is good. .5 = e^(5600*k) becomes: .5 = e^(5600*-.000123776) which becomes: .5 = .5 confirming the value for k is good. now that you have the value of k, you can solve the equation. if the tree has only 30% of the carbon left, then your equation of: F = P * e^(kt) becomes: .3 = 1*e^(k*t) which becomes: .3 = e^(-.000123776*t) take the log of both sides of this equation to get: log(.3) = log(e^(-.000123776*t)) which becomes: log(.3) = -.000123776*t*log(e) divide both sides of this equation by -.000123776*log(e) to get: t = log(.3) / (-.000123776*log(e)) solve for t to get: t = 9727.007327 years to confirm this is correct, substitute in the original equation to get: .3 = e^(-.000123776 * 9727.007327) which becomes: .3 = .3 the key is knowing the formula f = p*e^(kt) which I hope they gave you. without the use of the scientific constant of e, you would have had to at least know that the rate of growth or decay is exponential. solving this without the use of the formula f = p*e^(kt) is possible using exponential growth formula of: f = p*(1+x)^t this is a compounding formula where x is the rate of growth or decay. using this formula, you would get: f = p * (1+x)^t becomes: .5 = 1 * (1+x)^5600 which becomes: .5 = (1+x)^5600 take the 5600th root of both sides of this equation to get: (1+x) = .5^(1/5600) which becomes: (1+x) = .999876231 now that you know what (1+x) is equal to, you can solve the equation. equation of: f = p*(1+x)^t becomes: .3 = 1*(.999876231)^t which becomes; .3 = (.999876231)^t. take log of both sides of this equation to get: log(.3) = log(.999876231^t) which becomes: log(.3) = t*log(.999876231) divide both sides of this equation by log(.999876231) to get: t = log(.3) / log(.999876231) solve for t to get: t = 9727.007361 years. that's pretty close to t = 9727.007327 years which is what we got using the exponential formula of f = p*e^(kt) bottom line is: your answer is 9727.007327 years assuming exponential rate of growth or decay.
 Expressions-with-variables/250849: Zeno, Orb, Yurko and Sam are friends who live on neighboring space stations of the planet Krayon. They commute to school every day by space shuttle. Orb's space station is one half as far from Krayon as Zeno's space station. yurko travels as far as the total distance traveled by zeno and Orb, while sam travels three times the distance that Zeno travels. How many space miles does each friend travel to school, if the friends together travel a total of 888 space miles?1 solutions Answer 182717 by Theo(3458)   on 2009-12-17 06:34:26 (Show Source): You can put this solution on YOUR website!let: z = distance zeno has to travel o = distance that orb has to travel y = distance that yurko has to travel s = distance that sam has to travel. from the problem statement: o = z/2 y = z + o which is equivalent to z + z/2 s = 3z total distance they trafel is 888 space miles. this means that: z + o + y + s = 888 substituting all values in terms of z, we get: z + (z/2) + (z + z/2) + 3z = 888 simplifying by removing parentheses gets: z + z/2 + z + z/2 + 3z = 888 combining like terms gets: 6z = 888 dividing both sides of the equation by 6 gets: z = 148 o = z/2 = 148/2 = 74 y = z + o = 148 + 74 = 222 s = 3z = 444 148 + 74 + 222 + 444 = 888 confirming the answers are good. your answer is: zeno travels 148 space miles orb travels 74 space miles yurko travels 222 space miles sam travels 444 space miles
 Angles/250895: One angle is 4 degrees less then 3 times the other. What is the measure of the smaller angle? What is the measure of the other angle? How do you figure this out?1 solutions Answer 182707 by Theo(3458)   on 2009-12-17 04:52:49 (Show Source): You can put this solution on YOUR website!let y = one angle. let x = the other angle. your equation becomes: y = 3*x - 4 this means that y is 4 degrees less than 3 * x. you don't have enough information to solve for the measure of the angle. you have to be given the value of one or the other in order to solve that. you do have enough information to solve for the measure of the angle in terms of the measure of the other angle. using this formula, you can find the value of y given the value of x, but you have to be given the value of x. if you are not given the value of x, you can't find the value of y, because the value of y will vary based on the value of x. a graph of your equation shows the relationship. from this graph, you can see that the value of y is determined by the value of x. the graph is not accurate enough for you to see this directly, but if you use the formula to solve the equation for a given value of x, you will see that the value of y is in the general vicinity of that answer. for example, when x = 63, y = 3*63 - 4 = 185 degrees. I put a horizontal line at y = 185 to show you that the intersection of that line with the graph of the equation occurs at x = 63. drop a vertical line from that intersection and it will intersect the x-axis at approximately x = 63 if not right on. any deviation is due to the inaccuracy of the graphing method and or the display of the graph. the answer to your question is: the measure of the smaller angle is x. the measure of the larger angle is y. this was determined by relating the 2 angles through the use of a formula that was derived from the given statements about the relationship between the 2 angles.
 Surface-area/250457: I need to find the area of a trapezoid but the trapezoid isn't a isosceles trapezoid. The bases of this trapezoid is 10 and 24 and it has one leg that is 13 and the second leg is 15 I have to find the exact area of this problem1 solutions Answer 182375 by Theo(3458)   on 2009-12-16 05:55:51 (Show Source): You can put this solution on YOUR website!Use of Heron's formula may help. That gives you the area of a triangle given 3 sides of the triangle. The formula is: Area=sqrt(s(s-a)(s-b)(s-c)) s = sum of the sides of the triangle divided by 2. a,b,c are the sides of the triangle. your trapezoid looks like and is labeled like this: ``` A B xxxxxxxxxxxxxxxxxxxxxxxxxxxxx x x xx x x x x x x x x x x x x x x x x x x x x x x x x xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx C D E F ``` your trapezoid is ABFC you drop an altitude from A to D and from B to E. you have a rectangle at ABED you have 2 triangles at ADC and BEF CA is equal to 15 AB is equal to 10 BF is equal to 13 CF is equal to 24 DE is equal to 10 because it is the opposite side of AB in rectangle ABED. the length of your top base is AB = 10 the length of your bottom base is CF = 24 with DE representing 10 of it. if you remove the rectangle from the diagram, you are left with 2 triangles that connect together and form 1 triangle that has sides of 15, 13, and a base of 14. your modified diagram looks like this: ``` A x x xx x x x x x x x x x x x x x x x x x x xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx C D F ``` CA = 15 AF = 13 CF = 24 - 10 = 14 the sum of the perimeter is equal to 15 + 13 + 14 = 42 Heron's formula says the area of this triangle is equal to: Area=sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2 = 42/2 = 21 Formula becomes: Area = sqrt (21 * (21-15) * (21-13) * (21-14)) This resolves to: Area = 84 Now that we have the area of this triangle, we can find the height of the triangle which is also the height of the trapezoid. The area the triangle is equal to 1/2 * b * h. Since the area of the triangle is 84, this becomes: 84 = 1/2 * b * h Since the base of the triangle is 14, this becomes: 84 = 1/2 * 14 * h solve for h to get: h = 84 * 2 / 14 = 12 now that we have h we can solve for the area of the trapezoid. Area of the trapezoid = (1/2) * (b1+b2)*h. This equals to (1/2) * (10+24) * 12 = (1/2) * (34) * 12 Area of the trapezoid = 204 Here's a link to Heron's formula. http://mste.illinois.edu/dildine/heron/triarea.html I don't know if there's another way to solve this. I tried to find a way to get the height but was unsuccessful. It just didn't appear there was enough information provided.