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# Recent problems solved by 'stanbon'

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 , 3480..3509 , 3510..3539 , 3540..3569 , 3570..3599 , 3600..3629 , 3630..3659 , 3660..3689 , 3690..3719 , 3720..3749 , 3750..3779 , 3780..3809 , 3810..3839 , 3840..3869 , 3870..3899 , 3900..3929 , 3930..3959 , 3960..3989 , 3990..4019 , 4020..4049 , 4050..4079 , 4080..4109 , 4110..4139 , 4140..4169 , 4170..4199 , 4200..4229 , 4230..4259 , 4260..4289 , 4290..4319 , 4320..4349 , 4350..4379 , 4380..4409 , 4410..4439 , 4440..4469 , 4470..4499 , 4500..4529 , 4530..4559 , 4560..4589 , 4590..4619 , 4620..4649 , 4650..4679 , 4680..4709 , 4710..4739 , 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56970..56999 , 57000..57029 , 57030..57059 , 57060..57089 , 57090..57119 , 57120..57149 , 57150..57179 , 57180..57209 , 57210..57239, >>Next

 Equations/406336: 3 times x plus 7 = 4 times x1 solutions Answer 286762 by stanbon(57282)   on 2011-02-08 17:40:13 (Show Source): You can put this solution on YOUR website!3 times x plus 7 = 4 times x -------------- 3x + 7 = 4x Subtract 3x from both sides: x = 7 ====== Cheers, Stan H. ======
 Trigonometry-basics/406338: if sin data = 0, find all the possible values of A) cos data B) tan data C) sec data1 solutions Answer 286761 by stanbon(57282)   on 2011-02-08 17:38:10 (Show Source): You can put this solution on YOUR website!if sin data = 0, find all the possible values of A) cos data::: could be +1 of could be -1 ----- B) tan data:: is 0 ----- C) sec data:: could be +1 or could be -1 ===== Cheers, Stan H. =======
 real-numbers/406337: Worksheet titled: Making Connections with Factoring Connect factors with a line segment 1. X to the second power minus xy?1 solutions Answer 286760 by stanbon(57282)   on 2011-02-08 17:34:16 (Show Source): You can put this solution on YOUR website!X to the second power minus xy? ---- x^2-xy --- Common factor: x ---- = x(x-y) ============ Cheers, Stan H. ============
 Linear-systems/406332: I have a question about linear systems using substitution. How do solve: x=6-4y and 2x-3y=1 1 solutions Answer 286759 by stanbon(57282)   on 2011-02-08 17:31:14 (Show Source): You can put this solution on YOUR website!systems using substitution. How do solve: x=6-4y and 2x-3y=1 ---- Substitute for "x" and solve for "y": 2(6-4y)-3y = 1 12-8y-3y = 1 12-11y = 1 -11y = -11 y = 1 --- Now solve for "x": x = 6-4y x = 6-4*1 x = 2 ========== Cheers, Stan H. ===========
 Inverses/406318: The weight in pounds p of an olbject with a mass m of 25 kilograms is given by the equation p/m = 2.2. How many pounds does the object weigh?1 solutions Answer 286756 by stanbon(57282)   on 2011-02-08 17:28:06 (Show Source): You can put this solution on YOUR website!The weight in pounds p of an object with a mass m of 25 kilograms is given by the equation p/m = 2.2. How many pounds does the object weigh? ------------------------------------------- p/m = 2.2 p/25 = 2.2 p = 25*2.2 p = 55 lbs. ============================== Cheers, Stan H. ========
 Quadratic_Equations/406317: How do you find the equation of a graph of a parabola if the vertex is a decimal?1 solutions Answer 286750 by stanbon(57282)   on 2011-02-08 17:04:31 (Show Source): You can put this solution on YOUR website!Please post an example of your problem. Cheers, Stan H.
 Rational-functions/406313: Find the quadratic function f whose points on a graph are the vertex, which is (5,-4) and a second point that the parabola passes through is (7,-12). f(x)= ? How do I find the quadratic function?1 solutions Answer 286749 by stanbon(57282)   on 2011-02-08 17:02:54 (Show Source): You can put this solution on YOUR website!Find the quadratic function f whose points on a graph are the vertex, which is (5,-4) and a second point that the parabola passes through is (7,-12). f(x)= ? How do I find the quadratic function? ------ Use the vertex form: y = a(x-h)^2+k --- -12 = a(7-5)^2-4 ---- -12 = a(4)-4 4a = -8 a = -2 --- Equation: y = -2(x-5)^2-4 ====================== Cheers, Stan H. ===========
 Probability-and-statistics/406302: It has been found that IQ scores are normally distributes with a mean of 108 and a standard deviation of 16 points. A person must score in the upper 2% of the population in an IQ test to qualify for membership in Mensa. What is the minimum score that one has to achieve in order to qualify for Mensa?1 solutions Answer 286748 by stanbon(57282)   on 2011-02-08 16:58:41 (Show Source): You can put this solution on YOUR website!It has been found that IQ scores are normally distributes with a mean of 108 and a standard deviation of 16 points. A person must score in the upper 2% of the population in an IQ test to qualify for membership in Mensa. What is the minimum score that one has to achieve in order to qualify for Mensa? ---- Find the z-score with a right-tail of 0.02. invNorm(0.98) = 2.0537 --- Find the corresponding IQ score: x = zs + u x = 2.0537*16 + 108 x = 140.86 =============== Cheers, stan H.
 Probability-and-statistics/406307: In the game of blackjack as played in casinos, the dealer has the advantage as most of the players do not play very well. As a result, the probability that the average player wins a hand is about 40%. An average player decided to play 10 hands. What is the probability that he will win in all hands?1 solutions Answer 286745 by stanbon(57282)   on 2011-02-08 16:55:07 (Show Source): You can put this solution on YOUR website!In the game of blackjack as played in casinos, the dealer has the advantage as most of the players do not play very well. As a result, the probability that the average player wins a hand is about 40%. An average player decided to play 10 hands. What is the probability that he will win in all hands? --- P(win 10 hands out of 10) = 0.4^10 = 0.00010486 Cheers, Stan H. =========
 Age_Word_Problems/406304: Jorges father is 4 times as old as Jorge is. In 20 years his father will only be twice as old as Jorge. How old are Jorge and his father now?1 solutions Answer 286744 by stanbon(57282)   on 2011-02-08 16:52:28 (Show Source): You can put this solution on YOUR website!Jorges father is 4 times as old as Jorge is. In 20 years his father will only be twice as old as Jorge. How old are Jorge and his father now? ---- Equations: f = 4J (f+20) = 2(J+20) ------ Substitute for "f" and solve for "J": (4J+2) = 2J+40 2J = 38 J = 19 yrs (Jorge's age now) --- Solve for "f": f = 4J = 4*19 = 76 yrs (father's age now) ================== Cheers, Stan H. ==============
 Mixture_Word_Problems/406204: How many liters of a 40% alcohol solution must be mixed with a 60 L of a 80% solution to get a 50% solution?1 solutions Answer 286718 by stanbon(57282)   on 2011-02-08 14:35:29 (Show Source): You can put this solution on YOUR website! How many liters of a 40% alcohol solution must be mixed with a 60 L of a 80% solution to get a 50% solution? --------------------- Equation: alco + alco = alco 0.80*60 + 0.40x = 0.50(60+x) --- Multiply by 100 and solve for "x": 80*60 + 40x = 50*60 + 50x -10x = -30*60 x = (1/3)*60 x = 20L (amt. of 40% solution needed) ========================================= Cheers, Stan H. ==============
 test/406205: What is the average of the following numbers... 23/6, 17/4 , 7/2, 71 solutions Answer 286715 by stanbon(57282)   on 2011-02-08 14:31:39 (Show Source): You can put this solution on YOUR website!What is the average of the following numbers... 23/6, 17/4 , 7/2, 7 ----- Convert the denominators to 12 so you can add the fractions: --- (2*23)/12 ; (3*17)/12, (6*7)/12, (12*7)/12 ----- Add to get: 223/12 ----- Divide by the number of numbers to get the average: Ave = 223/48 =================== Cheers, Stan H. ===========
 Rate-of-work-word-problems/406194: Jane can paint a 12 x 12 foot wall in 3/4 hour. how long will it take her to paint antohter wall that is 15 x 16 feet?1 solutions Answer 286712 by stanbon(57282)   on 2011-02-08 14:26:32 (Show Source): You can put this solution on YOUR website!Jane can paint a 12 x 12 foot wall in 3/4 hour. how long will it take her to paint anothter wall that is 15 x 16 feet? ----- x/(15*16) = (3/4)/(12*12) ---- x = (15*16)(3/4)/(12*12) ---- x = 180/144 --- x = 1 1/4 hrs =================== Cheers, Stan H.
 Rational-functions/406199: When x = 2 and y = 3, z = 42. Write the function that models each of the following relationships. 1. z varies inversely with x and directly with y. 2. z varies jointly with x and y 3. z varies directly with x and inversely with the square of y 1 solutions Answer 286711 by stanbon(57282)   on 2011-02-08 14:22:10 (Show Source): You can put this solution on YOUR website!When x = 2 and y = 3, z = 42. Write the function that models each of the following relationships. 1. z varies inversely with x and directly with y. z = k*3/2 z = (3/2)k -------------------- 2. z varies jointly with x and y z = k*2*3 z = 6k ============== 3. z varies directly with x and inversely with the square of y z = k*x/y^2 z = k*2/9 z = (2/9)k ============== Cheers, Stan H.
 Functions/406201: evaluate the function rule for the given value. f(x)=3^x for x=-4 I would have tried the problem but i don't know how to start it. 1 solutions Answer 286710 by stanbon(57282)   on 2011-02-08 14:18:49 (Show Source): You can put this solution on YOUR website!evaluate the function rule for the given value. f(x)=3^x for x=-4 ----- f(-4) = 3^(-4) --- f(-4) = (1/3)^4 --- f(-4) = 1/81 ===================== Cheers, Stan H.
 Equations/406200: x=5y-3/y+2 solve for y1 solutions Answer 286709 by stanbon(57282)   on 2011-02-08 14:16:37 (Show Source): You can put this solution on YOUR website!x=5y-3/y+2 solve for y ----- Cross-multiply: xy+2x = 5y-3 ---- Get the y-terms together: xy-5y = -(2x+3) --- Factor out "y": y(x-5) = -(2x+3) --- y = -(2x+3)/(x-5) ======================== Cheers, Stan H.
 Polynomials-and-rational-expressions/406195: What are the factors of the polynomial: f(x)= x^3-10x^2+29x-20 ?1 solutions Answer 286708 by stanbon(57282)   on 2011-02-08 14:13:19 (Show Source): You can put this solution on YOUR website!What are the factors of the polynomial: f(x)= x^3-10x^2+29x-20 ----- The coefficients add up to zero so x=1 is a root, so (x-1) is one of the factors. --- Use synthetic division to get the quadratic factor: 1)....1....-10....29....-20 .......1.....-9....20...|..0 Quadratic factor: x^2-9x+20 Solve x^2-9x+20 = 0 (x-5)(x-4) = 0 ===================== Therefore: f(x) = (x-1)(x-4)(x-5) ======================================= Cheers, Stan H.
 Linear_Equations_And_Systems_Word_Problems/406187: Task: Write a linear equation to represent each situation. 11. My first test grade was 5 points less than my second test grade. This is what i wrote down.... S=F-5 What was the sum of my two test grades?1 solutions Answer 286705 by stanbon(57282)   on 2011-02-08 14:06:35 (Show Source): You can put this solution on YOUR website! My first test grade was 5 points less than my second test grade. F = S-5 --- What was the sum of my two test grades? Sum = first + second Sum = S-5+S Sum = 2S-5 ================ Cheers, Stan H. ================
 Equations/406096: Sorry for the wrong subject? I wasnt sure what this question would be listed as. Tell whether each sum is POSITIVE, NEGATIVE, OR ZERO. Explain your reasoning. n is positive and m is negativem n + (-m) is ___ ? n is positive and m is negative, -n + m is ____? |n| = |m|, n is positive, and m is negative. -n + (-m) is ____? n=m, and n and m are both the same sign. n + (-m) is ____? Thankyou! Please respond fast! :)1 solutions Answer 286670 by stanbon(57282)   on 2011-02-08 10:37:16 (Show Source): You can put this solution on YOUR website!Tell whether each sum is POSITIVE, NEGATIVE, OR ZERO. Explain your reasoning. n is positive and m is negativem n + (-m) is ___ ? If m is negative, -m is positive. positive + positive = POSITIVE ------------------------------------ n is positive and m is negative, -n + m is ____? If n is poitive -n is negative negative + negative = NEGATIVE ------------------------------------ |n| = |m|, n is positive, and m is negative. -n + (-m) is ____? If n is positve -n is negative If m is negative -m is positive Since |n|=|m|, m is the negative of n So, -n + (-m) = -n+(--n) = -n+n = ZERO -------------------------------------------- n=m, and n and m are both the same sign. n + (-m) is ____? ZERO =========================== Cheers, Stan H. ============
 Inequalities/406080: Hi i have two questions, one dealing with inequalities. Im only in pre algebra and i dont understand. Can you please answer this and maybe i will understand? Solve the following inequality: 2 < -8s Now for the second question its a word problem having to do with feet per second? Another person asked this same math question, but the answer was wrong! QUESTION : ACCORDING TO THE GUINNESS BOOK OF WORLD RECORDS, THE PEREGRINE FALCON HAS A RECORD DIVING SPEED OF 168 MILES PER HOUR. WRITE THIS SPEED PER SOCOND. Thankyou! Please respond quick. 1 solutions Answer 286658 by stanbon(57282)   on 2011-02-08 09:56:54 (Show Source): You can put this solution on YOUR website!Solve the following inequality: 2 < -8s Divide both sides by -8 to get: -1/4 < s Rewrite the result to get: s > -1/4 ======================== Now for the second question its a word problem having to do with feet per second? Another person asked this same math question, but the answer was wrong! QUESTION : -------------------------- ACCORDING TO THE GUINNESS BOOK OF WORLD RECORDS, THE PEREGRINE FALCON HAS A RECORD DIVING SPEED OF 168 MILES PER HOUR. WRITE THIS SPEED PER SECOND. ======================= [168 (miles/hour)] * [1 hr/3600 sec] --- = [168*1] / [1*3600](miles/sec) --- = 0.04667 miles/second ============================= Cheers, Stan H. ===========
 Quadratic_Equations/406079: Use the following quadratic equation to complete parts a and b below. You must show all work. x2 + 4x - 21 = 0 a. Solve by factoring. b. Solve using the quadratic formula. 1 solutions Answer 286656 by stanbon(57282)   on 2011-02-08 09:50:57 (Show Source): You can put this solution on YOUR website!x^2 + 4x - 21 = 0 a. Solve by factoring. x^2+7x-3x-21 = 0 x(x+7)-3(x+7) = 0 (x+7)(x-3) = 0 x = -7 or x = 3 ============================================= b. Solve using the quadratic formula. x = [-4 +- sqrt(16 - 4*1*-21)]/2 --- x = [-4 +- sqrt(16+84)]/2 --- x = [-4 +- 10]/2 --- x = [-2+- 5] --- x = -7 or x = 3 =================== Cheers, Stan H. ===============
 Numbers_Word_Problems/406078: one number is 10 less than a second number. twice the second number is 46 more than 4 times the first. Find the two numbers1 solutions Answer 286653 by stanbon(57282)   on 2011-02-08 09:44:38 (Show Source): You can put this solution on YOUR website!one number is 10 less than a second number. twice the second number is 46 more than 4 times the first. Find the two numbers ------------------------------ Let the numbers be "x" and "y": Equation: x = y-10 2y = 4x+46 --- Substitute for "x" and solve for "y": 2y = 4(y-10)+46 2y = 4y-40+46 -2y = 6 y = -3 ---- Solve for "x": x = y - 10 x = -3-10 x = -13 ---------- Cheers, Stan H. ============
 Probability-and-statistics/405932: Answer the following: (A) Find the binomial probability P(x = 5), where n = 12 and p = 0.60. (B) Set up, without solving, the binomial probability P(x is at most 5) using probability notation. (C) How would you find the normal approximation to the binomial probability P(x = 5) in part A? Please show how you would calculate µ and σ in the formula for the normal approximation to the binomial, and show the final formula you would use without going through all the calculations.1 solutions Answer 286641 by stanbon(57282)   on 2011-02-08 08:58:58 (Show Source): You can put this solution on YOUR website!(A) Find the binomial probability P(x = 5), where n = 12 and p = 0.60. Ans: 12C5(0.6)^5*(0.4)^7 = 0.1009 ----------------------------------------------- (B) Set up, without solving, the binomial probability P(x is at most 5) using probability notation. Ans: 12C0(0.6)^0*0.4^12+12C1(0.6)(0.4)^11+...+12C5(0.6)^5*(0.4)^7 ================================ (C) How would you find the normal approximation to the binomial probability P(x = 5) in part A? --- Find the z-values of 4.5 and 5.5 Find the probability z is between those two z-values. --- Please show how you would calculate µ and σ in the formula for the normal approximation to the binomial, and show the final formula you would use without going through all the calculations. u = np = 12*0.6 = 7.2 sigma = sqrt(npq) = sqrt(7.2*0.4) ======================================= Cheers, Stan H. ================
 Probability-and-statistics/405943: A marble is drawn at random from a bag containing 6 red marbles, 4 white marbles, and 5 blue marbles. Find the probability that it is: (a) red, (b) white, (c) blue, (d) not red, (e) red or white.1 solutions Answer 286640 by stanbon(57282)   on 2011-02-08 08:51:34 (Show Source): You can put this solution on YOUR website!A marble is drawn at random from a bag containing 6 red marbles, 4 white marbles, and 5 blue marbles. Find the probability that it is: (a) red,::6/15 (b) white,::4/15 (c) blue, (d) not red,::9/15 (e) red or white.::10/15 ============================= Cheers, Stan H.
 Probability-and-statistics/405991: A mini license plate for a toy car must consist of a vowel followed by two numbers. Each number must be a 1 or 6. Repetition of digits is permitted. Use the counting principle to determine the number of points in the sample space. Construct a tree diagram to represent this situation List the sample space. Determine the exact probability of creating a mini license plate with an A. Give solution exactly in reduced fraction form. 1 solutions Answer 286639 by stanbon(57282)   on 2011-02-08 08:48:51 (Show Source): You can put this solution on YOUR website!A mini license plate for a toy car must consist of a vowel followed by two numbers. VNN ---------------- Each number must be a 1 or 6. Repetition of digits is permitted. Use the counting principle to determine the number of points in the sample space. Ans: 5*2*2 = 20 ------------------------- Construct a tree diagram to represent this situation List the sample space. a11,a16,a61,a66,then the same letter e,i,o,u. ------ Determine the exact probability of creating a mini license plate with an A. Give solution exactly in reduced fraction form. Ans: 4/20 = 1/5 ====================== Cheers, Stan H. ================
 Rational-functions/406060: The one-to-one functions g and h are defined as follows. g={(-3,0),(0,9),(3,2),(8,-4)} h(x)=2x-3 find the following.. g^-1(0)=? h^-1(x)=? (h(h^-1))(-9)=? ( For this last one, I wasn't sure how to write out( h composite of h^-1) (-9). 1 solutions Answer 286638 by stanbon(57282)   on 2011-02-08 08:43:22 (Show Source): You can put this solution on YOUR website!The one-to-one functions g and h are defined as follows. g={(-3,0),(0,9),(3,2),(8,-4)} h(x)=2x-3 find the following.. g^-1(0)=-3 ----------------- h^-1(x)= (x+3)/2 ----------------------- (h(h^-1))(-9)= -9 ======================== Cheers, Stan H. ===============
 Coordinate-system/406030: find polar coordinates of each point (-2sqrt3,2)1 solutions Answer 286637 by stanbon(57282)   on 2011-02-08 08:39:44 (Show Source): You can put this solution on YOUR website!find polar coordinates of each point (-2sqrt3,2) ---- The point is in the 2nd quadrant. --- theta = tan^-1(2/(-2sqrt(3)) = tan^-1(-1/sqrt(3)) = (5pi/6) ---- r = sqrt[(2sqrt(3))^2 + 2^2] = sqrt[12+4] = 4 ---- polar coordinates: (4,5pi/6) =============================== Cheers, Stan H.
 Probability-and-statistics/405848: Find the value of z such that approximately 13.31% of the distribution lies between it and the mean.1 solutions Answer 286572 by stanbon(57282)   on 2011-02-07 21:06:44 (Show Source): You can put this solution on YOUR website!Find the value of z such that approximately 13.31% of the distribution lies between it and the mean. ---- Draw a normal curve and put the mean in the middle. --- Shade an area of 13.31% to the left of the mean. That has a left tale of 50-13.31 = 36.69% ----- Find the z-value with a left tail of 36.69%: --- z = invNorm(0.3669) = -0.3401 =============================== Similarly, if you shade the area on the right side of the mean you will get a z-value of +0.3401 ========================== Cheers, Stan H.
 Polynomials-and-rational-expressions/405830: college algebra - factoring quadratic quations i want to make sure i have my signs correct at the end x^2-4-12=0 (x-2) (x-6) x-2=0 x-6=0 +2 +2 +6 +6 x=2 x=6 Can i have both answers positive? 1 solutions Answer 286569 by stanbon(57282)   on 2011-02-07 20:59:15 (Show Source): You can put this solution on YOUR website!x^2-4x-12=0 --- (x-6)(x+2) = 0 --- x = 6 or x = -2 ===================== Cheers, Stan H. =========
 Equations/405840: Here is another one. Find an equation of the line having the given slope and containing the given point. m=4/5, (8, -8) The answer to the equation of the line is y= 4/5x-72/5. I did not get this answer and I don't know how they did. What am I missing? My answer was y=4/5x-40/5 Thank you for your assistance, Demi1 solutions Answer 286567 by stanbon(57282)   on 2011-02-07 20:54:20 (Show Source): You can put this solution on YOUR website!Find an equation of the line having the given slope and containing the given point. m=4/5, (8, -8) You have m,x,and y; solve for b: --- -8 = (4/5)8 + b --- (-40/5) = (32/8)+ b --- b = -72/5 --- Equation: y = (4/5)x-(72/5) ======================= Cheers, Stan H. ==========
 Equations/405838: der. instant rate of change 9x^2+4x-13... x=-7 f^1(-7) find the slope of the tangent i don't understand what the question is asking. 1 solutions Answer 286565 by stanbon(57282)   on 2011-02-07 20:50:47 (Show Source): You can put this solution on YOUR website!der. instant rate of change 9x^2+4x-13... x=-7 f^1(-7) find the slope of the tangent i don't understand what the question is asking. -------------------------- f'(x) = 18x+4 --- f'(-7) = 18(-7)+4 = -122 ---- The slope of the tangent at x = -7 is -122 ============================================= Cheers, Stan H. =============