Monday

February 8, 2016
Total # Posts: 32,839

**algebra**

When does .2 s + 1000 ≥ 2500 ? where s represents sales.
*February 3, 2016*

**MATHS**

n = 5m n = m+35 5m = m + 36 4m = 36 m = 9 n = 45 check: Is 45/9 = 5 ? YES Is 45-9 = 36 ? YES
*February 3, 2016*

**Algebra**

number on the floor --- g number on balconey --- b g + b = 1500 15f + 10b = 2100 ---> 3f + 2b = 420 solve for g and b hint: double the first and subtract from the 2nd
*February 3, 2016*

**Math**

y = 7x^2 + 9 dy/dx = 14x which is the slope of the tangent so when x = 5, slope = 70 and y = 7(25) + 9 = 184 so the tangent has a slope of 70 and goes through the point (5,184) equation of tangent: y - 184 = 70(x-5) y = 70x - 166 to get y1, replace x with 1 to get y2, replace ...
*February 3, 2016*

**Algebra**

welcome
*February 3, 2016*

**Algebra**

a = k/m^2 , where k is constant given: when a=2 m/s^2, m = 12 kg 2 = k/12^2 k = 288 a = 288/m^2 so when a = 3 3 = 288/m^2 3m^2 = 288 m^2 = 96 m= appr 9.8 the mass is appr 9.8 Kg or , just use a ratio 2/3 = m^2/12^2 3m^2 = 288 etc, same as above
*February 3, 2016*

**Math**

3y+5x=18 two step process to find the inverse of a linear function 1. interchange the x and y variables ---> 3x + 5y = 18 2. solve the new equation for y 5y = -3x + 18 y = (-3/5)x + 18/5 since the original was in standard form, we could have left the inverse in standard ...
*February 3, 2016*

**Math math help**

Ok, you are right, I mis-read the question. I was visualizing the arch above the road, with the road as an underpass. no big deal, easily fixed. so we can still use my equation etc. except we will make the road as y = 25 ft so at x = 0, the road = 25-20 or 5 ft above at x = 10...
*February 3, 2016*

**Math math help**

the distance form the arch to the roadway is the y value , the vertical, the up-and-down. at x = 0 , y = 20 ---> middle of road at x = 10, y = 18.856 it would be the same height at x = -10 of course it has to decrease, it would be highest in the middle.
*February 3, 2016*

**Math math help**

I was the one that gave you the above solution. What part of it don't you understand? Up to what point an you follow it ? I placed the ellipse on the gird so that the centre of the roadway is at (0,0) Since the far side of the road is at (30,0) wouldn't there be 10 ft ...
*February 3, 2016*

**algebra**

If I understand you correctly, you have the points (1,3), (2,-2), (3,x), (4,-24) and those points lie on a quadratic function graph. let the function by y = ax^2 + bx + c from (1,3) : a + b + c = 3 --- #1 from (2,-2) : 4a + 2b + c = -2 ---#2 from (4,-24) : 16a + 4b + c = -24...
*February 3, 2016*

**algebra**

simply convert e^x = 20.9 to the corresponding logarithmic equation. You MUST know how to do this. then simply use your calculator
*February 3, 2016*

**algebra**

let the unit digit be x let the tens digit by y the number is 10y + x the number reversed is 10x + y translate the English to Math (Math is a language) "divided by the sum of the digits the quotient is 7 and the remainder is 6 " (10y+x)/(x+y) = 7 + 6/(x+y) times (x+y...
*February 3, 2016*

**Algebra**

What is the symbol ã supposed to mean ? are we looking at ln √(x+4) = 1 ?? Two restrictions to consider: 1. we cannot take the square root of a negative, so x > -4 2. we can take the log of only a positive number. Show what steps you have taken.
*February 3, 2016*

**math**

I interpret a " 19/14 chain discount" to mean a 19% discount , followed by a 14% discount Manufacturer A's final price = (.81)(.86) = .6966 The equivalent single discount is 1-.6966 or 30.34 % other dealer's final price = (.76)(.88) = .6688 Sincle discount = ...
*February 3, 2016*

**Math**

5√3 y/(√10 + 2) what about it ?
*February 3, 2016*

**math**

how about y = (1/2)x^2 + 3 take a few points on the original and and the new one to confirm
*February 3, 2016*

**Algebra 1 college**

First of all, you always define your variables. I deduced from your work that you: let the amount of 12% solution used be T ml let the amount of 4% solution used be F ml clearly T + F = 40 is our first equation ( don't see that anywhere) then .12T + .04F = .06(40) times ...
*February 3, 2016*

**Math - help plz**

You didn't show your choices
*February 3, 2016*

**Math math**

Make a sketch of the semi-ellipse with centre (0,0) I assume you know the basic equation of an ellipse to be x^2/a^2 + y^2/b^2 = 1, where 2a is the major axis , and 2b is the minor axis in our case, a = 30, and b = 20 equation: x^2/900 + y^2/400 = 1 so we need the values of y ...
*February 3, 2016*

**Math**

If the terms are a GP, then 6/(3-x) = (7+5x)/6 36 = 21 + 15x - 7x - 5x^2 5x^2 - 8x +15 = 0 This quadratic has no real solution, thus the terms are not a GP in the set of real numbers. check your typing
*February 3, 2016*

**Permutation and combination**

I made up and ran a simple computer program and found this first one: 9^3 + 10^3 = 1^3 + 12^3 = 1729 I ran it only to 50^3 + 50^3, and found many of them I have shown the meaning of this with an example Notice that they up as duplications of pairs, too lazy to fix the program ...
*February 3, 2016*

**math**

16 = 4^2 36 = 6^2 64 = 8^2 81 = 9^2 100 = 10^2 I see no obvious pattern the 9^2 does not fit the otherwise obvious pattern.
*February 2, 2016*

**Math**

f(x) vs f(x) + 2 everything has just moved up vertically by 2 units so (3,-2) ---> (3,0)
*February 2, 2016*

**Math Please check**

I am not sure about it either, since you didn't say what has to be done with it. How did an expression suddenly turn into an equation? (1/8)17n x 8 , I assume that x means multiply = (1/8)(17n)(8) = 17n
*February 2, 2016*

**math ( factorising by grouping)**

I will do a couple , you do the rest following the same method b) 2ax-6bx+ay-3by = 2x(a - 3b) + y(a - 3b) you MUST have the same binomial in each of the terms = (a - 3b)(2x + y) or 2ax + ay - 6bx - 3by) = a(2x + y) - 3b(2x + y) = (2x+y)(a - 3b) showing that there is more than ...
*February 2, 2016*

**Algebra**

assuming the tank was empty, let the height of the water level be h ft (2)(3)(h) = 20 h = 20/6 = 3 1/3 ft or 3 ft, 4 inches or appr 3.33 ft As Ms Sue pointed out, the whole tank only holds 24 ft^3 how can you possible think that 480 ft^3 could be a reasonable answer ??
*February 2, 2016*

**math**

prob(red pen) = 50/200 = 1/4 prob(not red) = 3/4 prob(black pen) = 150/200) = 3/4 prob(not black) = 1/4 their sum = 3/4 + 1/4 = 1
*February 2, 2016*

**math**

functionA y = 9x + 4 ---->rate of change = 9 functionB y = 3x + 4 ----> rate of change = 3 so what do you think?
*February 1, 2016*

**math**

sin 30° = height/40 height = 40sin30 = ...
*February 1, 2016*

**algebra**

population = 25000(1.025)^2 = .... exponential
*February 1, 2016*

**math**

let the GP be a , ar, ar^2 , ar^3, ... the arithmetic (linear) sequence is a , a+d , a+2d , a + 3d , a+4d .. but 2nd term of GP= 3rd term of As ar = a + 2d * also for the AP a+3d = 10 ** and 5a + 10d = 60 a + 2d = 12 *** subtract (**) - (***) d = -2 <------------ then in...
*February 1, 2016*

**math**

I assume that 4% is an annual rate, so i = .04/12 = .00333... amount = 150( 1.00333...^312) - 1)/.00333.. = $82,094.77
*February 1, 2016*

**functions**

I will do c) follow the same steps for the others f(x) = (x-4)(x+3) f(-(x+1)) = (-(x+1) - 4)((-(x+1) + 3) = (-x-5)(-x+2) x-intercepts: -x-5 = 0 x = 5 or -x+2=0 x = 2
*February 1, 2016*

**Finance**

let her quarterly rate be i then 3126.28( 1 - (1+i)^-19)/i = 47500 1 - (1+i)^-19 = 15.193777i 1 - 1/(1+i)^19 = 15.193777i 1 - 15.193777i = 1/(1+i)^19 very nasty to solve, we used to do these using interpolation try some values: i = .01 --> PV = 53,853.33 too low i = .02 --&...
*February 1, 2016*

**maths**

sum(9) = (9/2)(2a + (n-1)d) (9/2)(2a + 8d) = 216 9(a + 4d) = 216 a + 4d = 24 term1 = a term3 = a+2d term7 = a + 6d they form a GP, so (a+2d)/a = (a+6d)/(a+2d) a^2 + 6ad = a^2 + 4d + 4d^2 2ad = 4d^2 a = 2d , assuming d ≠ 0 back into a+4d = 24 2d + 4d = 24 d = 4 a = 8 ...
*February 1, 2016*

**maths**

Easy to translate into Math ... let the first be x "the second number is twice the first " ---> 2x "the third number is 6times more than the second" ---> 2x+6 "the sum of 3 numbers is 81" --> x + 2x + 2x+6 = 81 solve for x , and sub into ...
*February 1, 2016*

**college pre-calc**

let the vertex be (0,15) then the equation is y = ax^2 + 15 the x-intercept is (8,0) 0 = 64a + 15 a = -15/64 y = (-15/64)x^2 + 15 so when x = 4.5 , assuming the truck goes down the middle of the road y = (-15/64)(4.5^2) + 15 = 10.25 but the truck is 12 ft tall, so it can't...
*January 31, 2016*

**Math**

2350(1.12)^n = 5000 1.12^n = 2.12766 take log of both sides n log 1.12 = log 2.12766 n = ....
*January 31, 2016*

**math - surface area and volume**

volume of sphere = (4/3)πr^3 = (4/3)π(9^3) = 972π cm^3 Increase in water volume = 972π radius of cylinder = 7.5 so π(7.5)^2 h = 972π 56.25h = 972 h = 17.28 cm So the water will rise 17.28 cm check my arithmetic
*January 31, 2016*

**surface area and volume**

surface area of similar objects varies directly with the square of their corresponding sides ratio of surface areas = 5^2 : 4^2 = 25 : 16
*January 31, 2016*

**Calculus and Vectors**

We don't just do your homework or assignment for you. You have posted 4 max/min problems without showing any steps or work or indicating where you difficulties lie. I have done two of them for you. Please show some work or steps before I do any more of them
*January 31, 2016*

**Algebra**

Proof-read you post and correct your wording so it can be understood.
*January 31, 2016*

**Calculus and Vectors**

let the radius of the can be r cm and its height be h cm given: πr^2 h = 1000 h = 1000/(πr^2) cost = .002(top+bottom) + .001(sleeve) = .002(2πr^2) + .001(2πrh) = .004πr^2 + .002πr(1000/πr^2 ) = .004πr^2 + .002(1000/r) = .002 [2πr^2...
*January 31, 2016*

**Calculus and Vectors**

let the width of the whole thing be x m and let the length be y m your sketch should show 3 widths and 2 lengths where 3x + 2y = 54 y = (54-3x)/2 = 27 - (3/2)x area = xy x(27 - (3/2)x) = 27x - (3/2)x^2 d(area)/dx = 27 - 3x = 0 for a max/min of area 3x = 27 x = 9 y = (54-27)/2...
*January 31, 2016*

**Calculus**

ln((3x-2)/(x-1))<0 using basic definition of logs .... ln(3x-2) - ln(x-1) < 0 by definition: x > 2/3 and x> 1 so x > 1 , before we even proceed further. but bob found that if we leave it as ln((3x-2)/(x-1)) < 0 we do get a solution of x < 1/2 thus x < 1...
*January 31, 2016*

**solid mensuration**

let the height be h √(4^2 + 3^2 + h^2) = √41 square both sides 16 + 9 + h^2 = 41 h^2 = 16 h = 4 volume = length x width x height = .....
*January 31, 2016*

**Maths**

number of boys --- 4x number of girls --- 3x given: 4x = 12 x = 3 number of girls = 3x = 9 or , by simple ratio 4/3 = 12/girls 4girls = 36 girls = 9
*January 31, 2016*

**Maths**

The way you typed it makes no sense. Did you mean he biked 66 km in 5 hrs ? I will assume that. distance covered at 12 km/h = 12x km distance covered at 16 km/h = 16y km 12x + 16y = 66 6x + 8y = 33 x + y = 5 y = 5-x plug this into the other equation: 6x + 8(5-x) = 33 6x + 40...
*January 31, 2016*

**maths**

tens digit --- x unit digit ---- 2x the actual number = 10x + 2x = 12x the number reversed = 10(2x) + x = 21x 12x + 21x = 99 33x = 99 x = 3 the original number was 63
*January 31, 2016*

**maths. It's urgent**

The graph of x^y + y^x =30 looks something like this http://www.wolframalpha.com/input/?i=plot+x%5Ey%2By%5Ex%3D30 is there an x-intercept? let y = 0 to get x^0 + 0^x = 30 1 + 0^x = 30 1 + 0 = 30 , false, there is no x-intercept same reason ---> no y-intercept but x+y = 15, ...
*January 31, 2016*

**number pattern**

term6 = ar^5 = -243 term3 = ar^2 = 72 divide: r^3 = -243/72 = -27/8 take cube root r = -3/2
*January 31, 2016*

**algebra**

use the rules of logs 2logx + log4 = 2 log(x^2) + log4 = 2 log (4x^2) = 2 4x^2 = 10^2 4x^2 = 100 x^2 = 25 x = ± 5, but we cannot take log(-5) so x = 5 test my answer with your calculator
*January 30, 2016*

**Mathematics**

1. x^3 - 16x ≤ 0 x(x^2 - 16) ≤ 0 x(x-4)(x+4) ≤ 0 critical values are -4, 0, 4 making a quick sketch of y = x^3 - 16x shows a standard cubic with x-intercepts of 0, ± 4 Where is the curve below the x-axis ? x ≤ -4 or 0 ≤ x ≤ 4 2. (5x/x...
*January 30, 2016*

**Math**

a reflection in the y-axis leaves the y's along but changes the x's to their opposites so P(2,-4) ---> P' (-2 , -4) repeat for the other points, then choose your answer
*January 30, 2016*

**math ratios help but explain**

I assume these two ratios are supposed to be the same. x:4 = 45:20 x/4 =45/20 times 4 x = 4(45/20) = 9
*January 30, 2016*

**algebra**

technicality here from (x+1)^2 we have (x+1)(x+1) so x = -1 or x = -1 same root shows up twice, so they call it a multiplicity of 2 your zeros are correct
*January 30, 2016*

**math**

or (4x - 1)(3x - 5) looks "nicer"
*January 30, 2016*

**math**

a+b-bc-ac = a - ac + b - bc = a(1 - c) + b(1 - c) = (a+b)(1-c) looks prettier
*January 30, 2016*

**Math**

1p + 2s + 2d - 2 = 21.08 but p = 11.7 and s = 11.7/3= 3.90 11.7 + 2(3.9) + 2 d - 2 = 21.08 solve for d, the cost of a drink
*January 30, 2016*

**Math**

i = .125/12 = .01041666.. balance after 1 month before payment = 7500(1.01041666.) payment = .05 of balance balance at end of month 1 = 7500(.95)(1.0104166) balance after month 2 before payment = 7500(.95)(1.0104166)^2 payment = .05 of that balance balance at end of month 2 = ...
*January 30, 2016*

**math- HELP!**

btw, The Fibonacci sequence was first NOT discovered during a study of rabbits.
*January 30, 2016*

**math**

Sum(33) = 6Sum(13) (33/2)(2a + 32d) = 6(13/2)(2a + 12d) but a = 2 33(4 + 32d) = 78(4 + 12d) sove for d
*January 30, 2016*

**College Algebra**

x^4-9x^2+20=0 (x^2 - 4)(x^2 - 5) = 0 x^2 = 4 or x^2 = 5 x = ± 2 or x = ± √5 All 4 roots are real, no complex roots
*January 30, 2016*

**math**

let B's amount be 6x let C's amount be 11x then A's amount = (3/5)(6x) = 18x/5 6x + 11x + 18x/5 = 1545 continue .....
*January 30, 2016*

**math**

Are we supposed to decipher what all that is ? Math is a precise language, and as such needs proper statements and presentation. Sample: let the width be x cm then the length is x+8 cm original area = x(x+8) cm^2 new width = 3x cm new length = x+8 - 9 = x-1 cm new area = 3x(x-...
*January 30, 2016*

**mathematics**

change the ratios to more manageable numbers 1/3 : 1/4 : 1/5 = 20 : 15 : 12 = 20x : 15x : 12x so 20x + 15x + 12x = 94 take over ...
*January 30, 2016*

**math**

Define the areas as 5x, 6x, and 8x respectively . given: flatC - flatA = 270 8x - 5x = 270 carry on ...
*January 30, 2016*

**math**

your last term is not clear is 1/12/16 = 1/(12/16) or = (1/12)/16 they are not the same we generally avoid writing fractions in such triple-decker form
*January 30, 2016*

**math**

What is 7(1.88) ?? $1.88 a gallon ? Time to get an up-to-date textbook
*January 30, 2016*

**Mathematics**

I will rename them as a , ar, and ar^2 given: ar = a+4 ar^2 = ar+9 ar^2 = a+4 + 9 = a + 13 but we also know that ar^2/ar = ar/a using the property of common ratio (a+13)/(a+4) = (a+4)/a a^2 + 13a = a^2 + 8a + 16 5a = 16 a = 16/5 common ratio = ar/a = (a+4)/a = (16/5 + 4)/(16/5...
*January 30, 2016*

**trigonometry**

Purists will argue that to prove the validity of an identity, that is, prove that the equation is true, by using the rules of equations itself is not valid.
*January 30, 2016*

**trigonometry**

I usually start these with changing everything to sines and cosines after I test them with some angle that I pick #1. The question should say: 1/(secx - tanx) = secx + tanx Those brackets are critical, the way you typed it, will not work LS = 1/(1/cosx - sinx/cosx) = 1/((1- ...
*January 30, 2016*

**math**

So if the content changed from 6/8 full to 3/8, the water was poured out not poured in. So the change is 3/8 of the capacity let that capacity be x litres (3/8)x = 3 x = 3(8/3) = 8 litres so 5 litres will be needed to make the jar full. btw, "how much liter will be needed...
*January 30, 2016*

**algebra**

let the unit digit be x let the tens digit be 4x original number: 10(4x) + x = 41x number reversed: 10x + 4x = 14x 41x + 14x = 110 55x = 110 x = 2 the original number is 84 try switching my definition and see what you get
*January 30, 2016*

**grade 12 math**

would 4 letters be enough? 26^4 = 459976 , no how about 5 letters ? 26^5 = 11,881,376 , not enough 26^6 = 308,915,776 That will do it. The string must be 6 letters long
*January 29, 2016*

**Calc 1 limits**

Look at your first post of this question. since we cannot extend the √ sign, we have to use brackets to show the end of the square root. e.g. is it t√(1+t) ???
*January 29, 2016*

**Calc 1**

I am going to go out on a limb and guess that you meant: lim ( √(1+t) - √(1-t) )/t , as t --->0 noting to factor here, you have to rationalize the numerator = lim ( √(1+t) - √(1-t) )/t * (√(1+t) + √(1-t))/(√(1+t) + √(1-t)) = ...
*January 29, 2016*

**Math**

you have the division backwards. you have 4 pizzas to be shared by 6 people so 4/6 or 2/3 of a pizza each check: what is 6 times 2/3 ?
*January 29, 2016*

**Limits calc 1**

OK THEN let's add up 1/( t√(1+t) - 1/t = (1 - √(1+t)/(t√(1+t) = (1 - √(1+t)/(t√(1+t) * (1 + √(1+t) / (1 + √(1+t) = (1 - 1 - t)/(t√(1+t)((1 + √(1+t) ) = -t/(t√(1+t)((1 + √(1+t) ) = -1/(√(1+t)((1 + &#...
*January 29, 2016*

**Limits calc 1**

Still not clear enough if it would be t√1 + t why would they put in that √1 ? it would simply become t(1) + t or 2t is it t√(1+t) ? there is a monumental difference should I guess at Limit (1/( t√(1+t) - 1/t ) , as x ---> 0 ???
*January 29, 2016*

**Limits calc 1**

Hard to make out what you mean without brackets for the denominators the way you typed it.... 1/t√1 is just 1/t, remember terms are separated by addition or subtraction so you simply have 1/t + t - 1/t = t I am sure that is not you meant.
*January 29, 2016*

**Math**

-15x+3 ≤ -3(4x-7) -15x + 3 ≤ -12x + 21 -3x ≤ 18 x ≥ -6
*January 29, 2016*

**Math**

-40 - (-55) = 15 yes
*January 29, 2016*

**Math**

cost = x -2 + 25 - 43 cost = x -20
*January 29, 2016*

**geometry**

I assume you mean that the dimensions are increased by 75% 2 inch --->2(1.75) = 3.5 3 inch ---> 3(1.75) = 5.25 your choice would have decreased the photo
*January 29, 2016*

**Math, pre-algebra**

since V = length(width)(height) 6x^3 - x^2 - 2x must be the product of 3 factors. the first one is obvious, a common factor of x = x(6x^2 - x - 2) and the quadratiac factors to ((2x+1)(3x-2) so which one fits ?
*January 29, 2016*

**Math-fractions**

No units were given for the 5/8 a book 5/8 of a foot wide is not very logical, so were the books only 5/8 inches? pretty skinny books. anyway 3 1/2 ft = 42 inches number of books = 42 ÷ 5/8 = 42(8/5) = 67.2 books ??? was expecting a whole number. Check your question.
*January 29, 2016*

**math**

franks rate = 1/3 Hanks rate = 1/x combined rate = 1/3 + 1/x = (x+3)/(3x) 1/( (x+3)/(3x) ) = 1 3x/(x+3) = 1 3x = x + 3 2x = 3 x = 1.5 It would take him 1.5 hours
*January 29, 2016*

**Math**

I will assume you have a typo in the last equation, and that 2c is 2z double the first: 4x + 4y + 2z = -10 ----> * subtract the end: x = -10 , well , that was lucky 4x + 4y + 2z = -10 x + 3y + 2c = 1 subtract them: 3x + y = -11 sub in x = -10 -30 + y = -11 y = 19 back in ...
*January 29, 2016*

**Math**

Use the rules of logs ln 5 + ln (2x) = 5 ln(5(2x)) = 5 ln (10x) = 5 10x = e^5 x = (1/10)e^5 = appr 14.84
*January 29, 2016*

**MATHS**

Of course ..... please ignore my reply, looks like I had a senior moment.
*January 29, 2016*

**MATHS**

number of hundreds --- h number of tens ------ t h = 6t 100h + 10t = 2440 100h + 60t = 2440 160t = 2440 t = 15.25 ----> not possible Your data is inconsistent.
*January 29, 2016*

**Math**

prob(3 hearts ) = C(13,3)/C(52/3) = 286/22100 = 11/850 expectation = (11/850)($50) = $0.647 prob(3 black) = C(26,3)/C(52,3) = 2600/22100 = 2/17 expectation = (2/7)($25) = $2.94 expected return on the game = $3.59 You were right on the $3.60 (difference is roundoff) agree with ...
*January 29, 2016*

**Math**

Your E(x) = $12.70 is correct your $1524 is correct I am not familiar with 'standard deviation' in this context
*January 29, 2016*

**Math**

We can't see your picture so you will have to describe what DE is
*January 29, 2016*

**Geometry**

nope ... distance = √((-2+1)^2 + (1-4)^2 + (3+2)^2 ) = √( 1 + 9 + 25) = √35 = appr 5.92 their choice is 5.9
*January 29, 2016*

**math**

Slopes of perpendicular lines are negative reciprocals of each other. One is + , the other is - . Here is a little trick for perpendicular lines If we have some line Ax + By + C = 0 then the line Bx -Ay + k = 0 is perpendicular, that is just switch the coefficients and make ...
*January 29, 2016*

**Calculus**

Step 1 in any limit question is to actually sub in the approach value into the expression. If you get 0/0 , you got some work ahead of you If you get (non-zero)/0 , there is no limit If you get a real number, that's it, you are done, that is your answer. So, let's try ...
*January 29, 2016*

**Math**

Even not remembering my physics, we can form the equation, since the relationship is given.. R = k L/d^2 25 = k(30)/.75^2 25 = k(30)(16/9) k = 9(25)/((16)(30)) = 15/32 R = (15/32)L/d^2 plug in the 2nd set of numbers 30 = (15/32) L /1.25^2 finish it up by solving for L
*January 29, 2016*