apply the property of the potency for those that have coefficients
[tex]b\cdot\log x=\log (x^b)[/tex]apply the product property and the quotient product to leave it as a single log
[tex]\begin{gathered} \log (a)+\log (b)=\log (a\cdot b) \\ \log (a)-\log (b)=\log (\frac{a}{b}) \end{gathered}[/tex]simplify the expression using this properties
[tex]\begin{gathered} \log _2(z)+2\log _2(x)+4\log (y)+12\log (x)-2\log _2(y) \\ \log _2(z)+\log _2(x^2)+\log (y^4)+\log (x^{12})-\log _2(y^2) \\ \log _2(x^2z)+\log (x^{12}y^4)-\log _2(y^2)^{} \\ \log _2(\frac{x^2z}{y^2})+\log (x^{12}y^4) \end{gathered}[/tex]Convert percent 26% of a number is what fraction of that number
Express 26% as a fraction:
26% = 26/100
[tex]\frac{26}{100}=\frac{13}{50}=0.26[/tex]Sarah wanted to catch Jim. However , although they started at the same time, Jim traveled at 80 km/h and Sarah traveled at 120 km/h . How much of a head start did Jim have if it took three hours for Sarah to catch him ?
Jim traveled at 80 km/h and Sarah traveled at 120 km/h
It took 3 hours for Sarah to catch Jim.
Let's find out how much distance both covered.
[tex]Jim\colon\; d=r\cdot t=80\cdot3=240\: km[/tex]So, Jim traveled 240 km
[tex]Sarah\colon\; d=r\cdot t=120\cdot3=360\: km[/tex]So, Sarah traveled 360 km
This means that Jim must have started 360 - 240 = 120 km ahead.
Therefore, Jim had a head start of 120 km
The force of the wind blowing on a window positioned at a right angle to the direction of the wind varies jointly as the area of the window and the square of the wind's speed. It is known that a wind of 30 miles per hour blowing on a window measuring 4 feet by 5 feet exerts a force of 150 pounds. During a storm with winds of 60 miles per hour, should hurricane shutters be placed on a window that measures 3 feet by 4 feet and is capable of withstanding 300 pounds of force?
We have the following:
The force of wind: F = 150 pounds
The square of the winds speed: V = 30 miles per hour
The area of the windows: A = 4*5 = 20 square feet
The formula is:
[tex]F=kAV^2[/tex]replacing:
[tex]\begin{gathered} 150=k\cdot20\cdot30^2 \\ k=\frac{150}{20\cdot900} \\ k=\frac{1}{120} \end{gathered}[/tex]now, the force of wind with 60 miles per hour and 12 (3*4) square feet
[tex]\begin{gathered} F=\frac{1}{120}\cdot12\cdot60^2 \\ F=360_{} \end{gathered}[/tex]The answer is 360 pounds
A person can join The Fitness Center for $50. A member can rent the tennis ball machine for $10 an hour. Write a linear function to model the relationship between the number of hours the machine is rented (x) and the total cost (y).
Answer: y=10x+50
Step-by-step explanation:
Which point satisfies both of the following inequalities? -3x + 5y< 15. 5x+y>-5
Explanation.
eWe are told to find the points that satisfy the systems of inequalities
The systems of equations are
[tex]undefined[/tex]y = -x - 2 y + 2 = -x Graph each system. Tell whether the system hasA.no solutionB.one solutionC. infinitely many solutionsD. Cannot determine
To graph each equation in the system, you can give it x-values, plug into the equations, and get values for Y.
Since a single line passes through two points, just take two values of x for each equation. So, for the first you have for example
*If x = 3
[tex]\begin{gathered} y=-x-2 \\ y=-3-2 \\ y=-5 \\ \text{ So} \\ (3,-5) \end{gathered}[/tex]*If x = -4
[tex]\begin{gathered} y=-x-2 \\ y=-(-4)-2 \\ y=4-2 \\ y=2 \\ \text{ So,} \\ (-4,2) \end{gathered}[/tex]For the second equation you have for example
*If x = 1
[tex]\begin{gathered} y+2=-x \\ y+2=-1 \\ y+2-2=-1-2 \\ y=-3 \\ \text{ So,} \\ (1,-3) \end{gathered}[/tex]*if x = -1
[tex]\begin{gathered} y+2=-(-1) \\ y+2=1 \\ y+2-2=1-2 \\ y=-1 \\ \text{ So,} \\ (-1,-1) \end{gathered}[/tex]Now, graphing the equations you have
As you can see, the lines associated with this system of equations overlap, that is, they share infinite solution points.
Therfore, the correct answer is C. infinitely many solutions.
find the unit price and round your answer to the nearest cent. you make $512.92 a week. if you work 36 hours find your hourly rate of pay
EXPLANATION
Given that we make $512.92 by week and we work 36 hours, we can apply the unitary method in order.
[tex]\text{hourly rate=}\frac{512.92\text{ dollars}}{36\text{ hours}}=14.25\text{ }\frac{dollars}{\text{hour}}[/tex]In conclusion, the hourly rate is 14.25 dollars.
Based on the graphs of f (x) and g(x), in which interval(s) are both functions increasing?Polynomial function f of x, which increases from the left and passes through the point negative 4 comma negative 4 and goes to a local maximum at negative 3 comma 0 and then goes back down through the point negative 2 comma negative 2 to a local minimum at the point negative 1 comma negative 4 and then goes back up through the point 0 comma 0 to the right, and a rational function g of x with one piece that increases from the left in quadrant 2 asymptotic to the line y equals 1 passing through the points negative 6 comma 2 and negative 2 comma 6 that is asymptotic to the line x equals negative 1 and then another piece that increases from the left in quadrant 3 asymptotic to the line x equals negative 1 passing through the point 0 comma negative 4 and 4 comma 0 that is asymptotic to the line y equals 1(–°, –3) ∪ (–1, °)(–°, –3) ∪ (4, °)(–°, –3)(–°, °)
Increasing intervals for f(x) and g(x); A function is incrasing when the y-value increases as the x-value increases.
f(x) and g(x) increases in two intervals (x-intervals):
From negative infinite to -3
From -1 to infinite
[tex](-\infty,-3)\cup(-1,\infty)[/tex]Both intervals are increasing through (-∞,-3) U (-1,∞)
(5 points each for a and b) A bacteria colony starts with 20 bacteria andgrows continuously at a rate of 28% per hour.a. How long will it take for the colony to:i. Double its size?ii. Reach 500,000 bacteria?b. How many bacteria will there be in:i. 3 hours?ii. 3.5 days?
Given:
The initial population of bacteria, I = 20
Growth rate, r = 28%
Explanation:
a) To find: The time
i) Double its size
Using the formula,
[tex]F=I(1+r)^t,\text{ Where I denotes initial and F denotes Final size.}[/tex]On substitution we get,
[tex]\begin{gathered} 40=20(1+0.28)^t \\ 1.28^t=\frac{40}{20} \\ t=\log _{1.28}2 \\ t=2.81 \end{gathered}[/tex]Thus, the answer is 2.81 hours.
ii) To reach 500000 bacteria:
[tex]\begin{gathered} 500000=20(1+0.28)^t \\ 1.28^t=\frac{500000}{20} \\ t=\log _{1.28}(25000) \\ t=41.02\text{ hours} \end{gathered}[/tex]Thus, the answer is 41.02 hours.
b) To find the bacteria size:
i) In 3 hours,
[tex]\begin{gathered} F=20(1+0.28)^3 \\ =41.94304 \\ \approx42 \end{gathered}[/tex]Thus, the size of bacteria in 3 hours is 42.
i) In 3.5 days,
That is, 84 hours
[tex]\begin{gathered} F=20(1+0.28)^{84} \\ =20261306488.67 \\ \approx20261306489 \end{gathered}[/tex]Thus, the size of bacteria in 3.5 days is 20261306489.
Find greatest common factor for each group,factor completely and find real roots
SOLUTION
Write out the polynomial given
The first group of the expresion is
[tex]\begin{gathered} 3x^3+4x^2 \\ \text{Then the GCE is } \\ x^2(\frac{3x^3}{x^2}+\frac{4x^2}{x^2}) \\ \text{GCE}=x^2 \end{gathered}[/tex]GCE is x²
For the second group, we have
[tex]\begin{gathered} 75x+100 \\ \text{GCE}=25(\frac{75x}{25}+\frac{100}{25}) \\ \text{GCE}=25 \end{gathered}[/tex]The GCE for the secod group is 25
To factorise completely, we have
[tex]\begin{gathered} 3x^3+4x^2+75x+100 \\ \\ x^2(\frac{3x^3}{x^2}+\frac{4x^2}{x^2})+25(\frac{75x}{25}+\frac{100}{25}) \end{gathered}[/tex]Then by simplification, we have
[tex]\begin{gathered} x^2(3x+4)+25(3x+4) \\ \text{Then, we factor completely to get} \\ (3x+4)(x^2+25) \end{gathered}[/tex]Then factors are (3x +4)(x²+ 25)
To find the real root, we equate each of the factors to zero, hence
[tex]\begin{gathered} (3x+4)(x^2+25)=0 \\ \text{Then} \\ 3x+4=0orx^2+25=0 \\ 3x=-40rx^2=-25 \\ \end{gathered}[/tex]Thus
[tex]\begin{gathered} \frac{3x}{3}=-\frac{4}{3} \\ x=-\frac{4}{3}\text{ is a real root } \\ or\text{ } \\ x^2=-25 \\ \text{take square root} \\ x=\pm_{}\sqrt[]{-25}\text{ not a real root} \end{gathered}[/tex]Therefore, since the root of -25 is a complex number,
The only real root is x = -4/3
Complete the square to findthe vertex of this parabola.x² - 2x + y - 4 = 0([?], [ ])
Given:
[tex]x^2-2x+y-4=0[/tex]Let's complete the square to find the vertex of the parabola.
To solve first move all terms not containing y to the right side of the equation:
[tex]y=-x^2+2x+4[/tex]Now, take the vertex form of a parabola:
[tex]y=a(x-h)^2+k[/tex]Apply the standard form of a parabola:
[tex]\begin{gathered} ax^2+bx+c \\ \\ -x^2+2x+4 \end{gathered}[/tex]Thus, we have:
a = -1
b = 2
c = 4
Now, to find the value of h, we have:
[tex]\begin{gathered} h=-\frac{b}{2a} \\ \\ h=-\frac{2}{2(-1)} \\ \\ h=-\frac{2}{-2} \\ \\ h=1 \end{gathered}[/tex]To find the value of k, we have:
[tex]\begin{gathered} k=c-\frac{b^2}{4a} \\ \\ k=4-\frac{2^2}{4(-1)} \\ \\ k=4-\frac{4}{-4} \\ \\ k=4+1 \\ \\ k=5 \end{gathered}[/tex]We have the values:
h = 1
k = 5
The vertex of the parabola is:
(h, k) ==> (1, 5)
ANSWER:
(1, 5)
I just need the answer for this, no process or explanation needed, thank you!
1) Considering that the initial amount of Carbon 14 is 40% (0.4) and each year "t" is given in whole numbers. And we were told the amount of k we can insert into the equation:
[tex]\begin{gathered} N=N_0e^{-kt} \\ 0.6=\left(1\right)e^{-0.0001*t} \\ e^{\left\{-0.0001t\right\}}=0.6 \\ lne^{\left\{-0.0001t\right\}}=ln\left(0.6\right) \\ -0.0001t=ln\left(0.6\right) \\ t=5108.25623 \\ t\approx5108years \end{gathered}[/tex]Note that the initial value is 1 and the last one is 0.6 (40% less).
A person collected $1,400 on a loan of $1,200 they made 7 years ago. If the person charged simple interest, what was the rate of interest?
Solution:
The formula that we can apply in this case is the following:
[tex]r\text{ = (}\frac{1}{t})(\frac{A}{P}-1)[/tex]now, solving we get:
[tex]r\text{ = (}\frac{1}{7})(\frac{1400}{1200}-1)=\text{ }0.02380952[/tex]if we convert this amount into a percentage we get the final answer:
[tex]0.02380952\text{ x 100\% = }2.381[/tex]then, the correct answer is:
2.381% per year
Identify any misrepresentation issues in the given graph. ▪︎The horizontal axis scale is not appropriate. ▪︎The horizontal axis ticks are not placed correctly.▪︎ The vertical axis scale is not appropriate. ▪︎The vertical axis ticks are not placed correctly. ▪︎The axis labeling is not complete. ▪︎There are distracting visual effects. ▪︎The graph is designed appropriately.
Looking at the graph, we can see that the x-axis has a lot of unused values, it goes until 50 but the graph only goes until around 25, so the horizontal axis scale is not appropriate.
Also, the x-axis does not have a label, so The axis labelling is not complete.
Last, this is a good graph to represent the temperature over the time, so the graph is designed appropriately.
Find the measure of base of the following parallelogram shown below.Area =10.92 cm?2.6 cmAnswer:cm
The area of a paralllelogram can be found by multiplying its base with its height. In this problem we were given the area and the height, therefore we can solve for the base as shown below.
[tex]\begin{gathered} Area=base\cdot height \\ ase=\frac{Area}{height} \\ base=\frac{10.92}{2.6} \\ base=4.2\text{ cm} \end{gathered}[/tex]The base of the parallelogram is 4.2 cm
The doctor orders 3000 mL D5RL to run at 300 ml/hr. How long will this IV infusion run
we know that
the infusion run at 300 ml/hr
so
Applying proportion
Find out how long for 3,000 ml
Modeling System of Equations Per 2
Based on the given information, you can write the following equations for the costs:
y1 = 35x + 75
y2 = 38x
If the cost is the same for both companies, you have:
35x + 75 = 38x
you can solve the previous equation for x to determine the number of people:
35x + 75 = 38x subtract 35x both sides
75 = 38x - 35x
75 = 3x divide by 3 both sides
75/3 = x
25 = x
Hence, the number of people is 25
The number a exceeds the number b by 50% by what percent is the number b smaller than the number a
Number a exceeds the number b by 50%:
[tex]\begin{gathered} a=b+0.5b \\ a=1.5b=\frac{3}{2}b \\ b=\frac{2}{3}a \end{gathered}[/tex]That means, 1/3 smaller=33, 1/3% smaller or 33% smaller.
Given the triangle congruence statement ΔUVW≅ΔABC and the triangles below, mark each of the triangles appropriately for corresponding angles and sides. Then create a list of congruent sides and angles using the tableCorresponding SidesCorresponding Angles
Given:
[tex]\Delta UVW\cong\Delta ABC[/tex]Corresponding sides are:
[tex]\begin{gathered} UV\cong AB \\ VW\cong BC \\ UW\cong AC \end{gathered}[/tex]Corresponding angles are:
[tex]\begin{gathered} \angle U\cong\angle A \\ \angle V=\angle B \\ \angle W=\angle C \end{gathered}[/tex]WhT is the slop of (0,3)
Step 1;
Write the coordinates of the given points
I don't understand how to do a certain equation i have no clue what its called
Answer:
y =24 -x
Explanation:
Mr Ledger has 24 donuts and if he uses x donuts then the amount he will have left will be
[tex]24-x[/tex]and since we are calling this amount left y, we can say
[tex]y=24-x[/tex]which is our answer!
When Mr Ledger uses 18 donuts, the number left will be
[tex]\begin{gathered} y=24-18, \\ y=6. \end{gathered}[/tex]And when Mr Ledger uses 22 donuts, we will have left
[tex]\begin{gathered} y=24-22, \\ y=2. \end{gathered}[/tex]donuts.
The graph of the two points is given below.
The volume of a gas, such as helium or air, varies inversely with the pressure on it. If the volume of air is 325 cubic inches under a pressure of 11 psi, what pressure has to be applied to decrease the volume to 143 cubic inches?
ANSWER
The pressure is 25 psi
STEP-BY-STEP EXPLANATION:
From the question provided, you can see that the relationship between the volume of a gas and the pressure is an inverse relationship.
The volume of a gas varies inversely with the pressure on it
This implies that as the volume of the gas increases the pressure of the gas decreases and vice versa.
The next thing is to assign variables
Let the volume of the gas be V
Let the pressure of the gas be P
Mathematically, this can be represented as
[tex]\begin{gathered} V\text{ }\propto\text{ }\frac{1}{P} \\ \text{Introduce a proportionality constant K} \\ V\text{ = }\frac{K}{P} \\ \text{Cross multiply} \\ K\text{ = VP -------- equation 1} \\ \end{gathered}[/tex]The next step is to find the value of K from the given information in the question
• Volume = 325 cubic inches
,• Pressure = 11 psi
Recall that, K = VP
K = 325 * 11
K = 3,575
Since you have gotten the value of K, then, you can now find your pressure when the volume Is 143 cubic inches
[tex]\begin{gathered} V=143inches^3 \\ K\text{ = 3,575} \\ K\text{ = VP} \\ \text{Divide both sides by V} \\ \frac{K}{V}\text{ = }\frac{VP}{V} \\ P\text{ = }\frac{K}{V} \\ P\text{ = }\frac{3575}{143} \\ P\text{ = 25 ps}i. \end{gathered}[/tex]Hence, the pressure is 25 psi
Solve the equation below for x. log(5x) + log(2x) = 1 O A. x = 10/7 O B. x = 1; x = -1 O c. x= 1 O D. There is no solution.
So we need to solve the following equation:
[tex]\log (5x)+\log (2x)=1[/tex]There are a few properties of logarithmic functions that we should remember. First, the logarithm of a negative number doesn't exist which means that x must be a positive number. Second, the addition of logarithms meets the following property:
[tex]\log (a)+\log (b)=\log (a\cdot b)[/tex]If we apply this to our equation we get:
[tex]\log (5x)+\log (2x)=\log (5x\cdot2x)=\log (10x^2)=1[/tex]Now we can pass the logarithm to the right side of the equation:
[tex]\begin{gathered} \log (10x^2)=1 \\ 10x^2=10^1=10 \\ 10x^2=10 \\ x^2=1 \end{gathered}[/tex]There are two possible solutions for x^2=1. These are x=1 and x=-1, however as I stated before x can't be a negative number which means that the solution of the equation is:
[tex]x=1[/tex]Then option C is the correct one.
Use a net to find the surface area of the prism. The surface area of the prism is ___cm² (Simplify your answer.)
Answer:
1,417 cm²
Explanation:
The net of the prism is attached below:
The surface area of the prism is the area of each of the triangles.
[tex]\begin{gathered} \text{Surface Area=}(13\times32)+(32\times6.5)+(13\times32)+(32\times6.5)+(13\times6.5)+(13\times6.5) \\ =416+208+416+208+84.5+84.5 \\ =1417\operatorname{cm}^2 \end{gathered}[/tex]The surface area of the prism is 1,417 cm².
a railroad tracks can be determined using the following graph. Several different rosdways are in the same region as the railroad. Part B: A turnpikes route is determined by the equation y=1/3x^2. Prove algebraically how many intwrsections there will be between the railroad abd the turnpike,showing all necessary work
Given
[tex]y=\frac{1}{3x^2}[/tex]2x+3y=18
Find
Prove algebraically how many intwrsections there will be between the railroad
Explanation
The graph of 2x+3y=18 is as the picture
2x+3y=18
when x=0, 0+3y=18 => y=6 =>(0,6)
when y=0, 2x+0=18 => x=9 => (9,0)
The intersection between the railroad and the highway is 0 because the graph of the railroad and the graph of the highway are parallel, that means they have no intersection
(b)
Assume the railroad can be found using the equation y=3/2x+b
when x=0 => y=8
[tex]\begin{gathered} \frac{1}{3}x^2=\frac{3}{2}x+8 \\ 2x^2-9x-48=0 \\ D=9^2-4(2)(-48)=465 \\ =>D>0 \\ \frac{1}{3}x^2=\frac{3}{2}x+8 \end{gathered}[/tex]has two roots, and there are 2 intersections
Final Answer
(a) No intersection
(b) Two intersections
Suppose that you borrow $14,000 for five years at 6% toward the purchase of a car. Find the monthly payment and the total interest for the loan.
We have to calculate the monthly payments (number of subperiods per year n = 12) for a loan of $14,000 (P = 14000) for five years (t = 5) at an interest rate of 6% (r = 0.06).
We can use the annuity formula to calculate the monthly payment (PMT) as:
[tex]\begin{gathered} \text{PMT}=\frac{P(\frac{r}{n})}{\lbrack1-(1+\frac{r}{n})^{-nt}\rbrack} \\ \text{PMT}=\frac{14000\cdot(\frac{0.06}{12})}{\lbrack1-(1+\frac{0.06}{12})^{-12\cdot5}\rbrack} \\ \text{PMT}=\frac{14000\cdot0.005}{\lbrack1-1.005^{-60}\rbrack} \\ \text{PMT}\approx\frac{14000\cdot0.005}{1-0.74137} \\ \text{PMT}\approx\frac{70}{0.25863} \\ \text{PMT}\approx270.66 \end{gathered}[/tex]Answer: the monthly payments will be $270.66
-Determine whether each relation is a function. Explain your answerA. {(7,4),(6,3),(5,2)}B. {(15,0),(15,-2)}C. {(0,1),(2,1),(0,3)}
A. it is a fuction because for every number in the first position of the pairs it is one and only one second position number.
B. it's not a function because the number 15 has two different pairs
C. it's not a function because zero has two different pairs, 1 and 3.
write the number in standard notation:3.92x10^7
We will write it as follows:
[tex]3.92\cdot10^7=39200000[/tex]Given a student has a dog, what is the probability that a student also has a cat?62.9%57.1%41.8%36.3%
The given problem is a conditional probability problem.
Probability that a student has a cat given that he/she has a dog is represented as:
[tex]Pr(C|D)=\frac{Pr(CnD)}{Pr(D)}[/tex][tex]\begin{gathered} \text{CnD}=16 \\ \text{Sample space=28+16+24}=68 \end{gathered}[/tex]Thus,
[tex]Pr(\text{CnD)}=\frac{16}{68}=0.2353[/tex][tex]\begin{gathered} Number\text{ of dogs only=28} \\ Pr(D)=\frac{28}{68}=0.4117 \end{gathered}[/tex]Therefore,
[tex]undefined[/tex]If the fish tanks dimension are 60 by 15 by 34 and its is completely empty, what volume of water is needed to fill three fourths of the aquarium? Please help what would the volume if you only filled 3/4 of the tank
First let's find the volume of the fish tank. Given that the dimensions are 60 by 15 by 34, then:
[tex]V=(60)(15)(34)=30600[/tex]we have that the total volume of the fish tank is 30600 u³. But we only want to know how much is 3/4 of the total volume, then:
[tex](30600)(\frac{3}{4})=22950[/tex]therefore, to fill three fourths of the aquarium we will need 22950 u³ of water