Tutor profile: Kayla S.
In your chemistry lab, the lab instructions call for 100 mL of 3M NaOH. Unfortunately, the lab stock solution is 50M NaOH. What steps do you need to take to make 100 mL of 3M NaOH?
To figure out how to make 100 mL of 3M NaOH from a 50M stock solution of NaOH, you need to use the equation M1V1 = M2V2. M stands for molarity (or concentration of any kind as long as both M1 and M2 are in the same units) and V stands for volume. Lets say that solution 1 is our goal solution (the 100mL 3M NaOH) and solution 2 is the stock solution. You know that you want a volume of 100 mL (V1) and concentration of 3M (M1) and the stock solution has a concentration of 50M (M2). Now we have to solve for the unknown variable V2 M1V1 = M2V2 (100mL)(3M) = (50M)V2 V2 = 6 mL But you cannot stop at just 6mL. What does that 6mL mean? It means that you must take 6mL of the stock solution (50M NaOH) and mix it with enough water to make 100mL of total solution (100mL - 6mL = 94 mL). So in order to make this solution we need for the lab, we need to mix 6mL of the 50M NaOH stock solution with 94mL of water.
Two pea plants are bred to produce offspring. For this question, all traits and genes are either dominant or recessive (no incomplete dominance or other forms of inheritance). Two genes will be explored: color (GG, Gg, gg; the color green is dominant and yellow is recessive) and height (TT, Tt, tt; the tall phenotype is dominant and short phenotype is recessive). One plant was homozygous dominant for color and homozygous recessive for height The other plant was heterozygous for both traits. At what frequency in the progeny will we see each phenotype and genotype?
To answer this problem, we must first determine the genotypes for both parents. Homozygous dominant for color (green) means the genotype of parent 1 is two dominant alleles (GG) and homozygous recesive for height means it has two recessive alleles (tt) Parent 1: GGtt Heterozygous for both traits means that parent 2 shows dominant phenotypes for both traits (green, tall) but has two different alleles (Gg, Tt) Parent 2: GgTt Now we have to cross the parents using Punnett squares. I like to do this separately for each trait: GG x Gg will yield 1/2 GG and 1/2 Gg (all offspring will be green phenotypically but half will be homozygous dominant and half will be heterozygous) tt x Tt will yield 1/2 Tt and 1/2 tt (1/2 of offspsring will be tall with a heterozygous genotype and 1/2 of the offspring will be short with homozygous recessive genotype) Then we can combine these frequencies to find the overall frequencies for each genotype and phenotype: Genotypes: 1/2 GG x 1/2 Tt = 1/4 GGTt 1/2 GG x 1/2 tt = 1/4 GGtt 1/2 Gg x 1/2 Tt = 1/4 GgTt 1/2 Gg x 1/2 tt = 1/4 Ggtt Phenotypes: Green, tall = 1/4 GGTt + 1/4 GgTt = 1/2 Green, short = 1/4 GGtt + 1/4 Ggtt = 1/2 yellow, tall = none yellow, short = none
The border of two properties can be represented by the following equations: Property A: y = 1/2x - 1 Property B: y = -3x^2 + 5x +1 Do these properties overlap? If so use principles of calculus to find the area between the overlapping properties.
The first step is to figure out if the functions overlap. This can be done a number of ways. You can graph them and see if they overlap or you can set them equal to each other and see if they cross one another. In this case the equations cross twice. Where they cross is also important and you can get these points from the graph or from setting the equations equal to each other and solving for x. (at x = 0, -3/2) Now that we know they do overlap, we have to find out by how much. To do this we can take the area under one curve and subtract the area under the other. Finding area under a curve is done using an integral. Property B is higher up on the graph than A so we will subtract A from B. Area = ∫ Property B - Property A (note this will be from a lower limit of -3/2 to an upper limit of 0 as these are the two points where the equations meet) Area = ∫ (-3x^2 + 5x +1) - (1/2x - 1) dx I like to simplify like terms first which looks like this: -3x^2 + 5x + 1 - 1/2x + 1 = -3x^2 + 9/2x + 2 Area = ∫(-3x^2 + 9/2x + 2)dx Then we can take the integral of each individual piece which will get us: -x^3 + 9/4x^2 + 2x Then we have to complete the integral by taking f(-3/2) - f(0) (between the two points of intersection) [-(-3/2)^3 + 9/4(-3/2)^2 + 2(-3/2)] - [-(0)^3 + 9/4(0)^2 + 2(0)] [87/16] -  = 87/16 This means the overlapping area of the properties is 87/16 units (no units specified)
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