We have the following situation regarding the growth of bacteria in a culture:
• The given initial population of bacteria is 27,000
,• After 5 hours, the population increases to 105,000.
Now, we need to find the moment when that population is one million if:
• The population increases linearly with time
,• The population increases exponentially with time
To find the time in both situations, we can proceed as follows:
Finding the moment when the population is one million if it increases linearly with time1. We need to find the equation of the line that passes the following two points:
• t = 0, population = 27,000
,• t = 5, population = 105,000
2. Then the points are:
[tex]\begin{gathered} (0,27000)\rightarrow x_1=0,y_1=27000 \\ (5,105000)\rightarrow x_2=5,y_2=105000 \\ \end{gathered}[/tex]3. Now, we can use the two-point form of the line equation:
[tex]\begin{gathered} y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1) \\ \\ y-27000=\frac{105000-27000}{5-0}(x-0) \\ \\ y-27000=\frac{78000}{5}x=15600x \\ \\ y=15600x+27000\rightarrow\text{ This is the line equation we were finding.} \end{gathered}[/tex]4. We can see that the population is given by y. Then if y = 1,000,000, then we need to solve the equation for x as follows:
[tex]\begin{gathered} 1000000=15600x+27000 \\ \\ 1000000-27000=15600x \\ \\ \frac{(1000000-27000)}{15600}=x \\ \\ x=62.3717948718\text{ hours} \\ \\ x\approx62.3718\text{ hours} \end{gathered}[/tex]Therefore, if the population increases linearly with time, the number of bacteria will be one million around 62.3718 hours.
Finding the moment when the population is one million if it increases exponentially with time1. In this case, we also need to find the equation that will give us the time when the number of bacteria is one million. However, since the equation will be exponential, we have:
[tex]\begin{gathered} y=a(1+r)^x \\ \\ a\rightarrow\text{ initial value} \\ \\ x\rightarrow\text{ number of time intervals that have passed.} \\ \\ (1+r)=b\text{ }\rightarrow\text{the growth ratio, and }r\rightarrow\text{ the growth rate.} \end{gathered}[/tex]2. Now, we can write it as follows:
[tex]\begin{gathered} a=27000 \\ \\ x=5\rightarrow y=105000 \\ \\ \text{ Then we have:} \\ \\ 105000=27000(b)^5 \\ \end{gathered}[/tex]3. We can find b as follows (the growth factor):
[tex]\begin{gathered} \frac{105000}{27000}=b^5 \\ \\ \text{ We can use the 5th root to obtain the growth factor. Then we have:} \\ \\ \sqrt[5]{\frac{105000}{27000}}=\sqrt[5]{b^5} \\ \\ b=1.31209447568 \end{gathered}[/tex]4. Then the exponential equation will be of the form:
[tex]\begin{gathered} y=27000(1.31209447568)^x \\ \\ \text{ To check the equation, we have that when x = 5, then we have:} \\ \\ y=27000(1.31209447568)^5=105000 \end{gathered}[/tex]5. Now, to find the time when the number of bacteria is one million, we can proceed as follows:
[tex]\begin{gathered} 1000000=27000(1.31209447568)^x \\ \\ \frac{1000000}{27000}=1.31209447568^x \end{gathered}[/tex]6. Finally, we need to apply the logarithm to both sides of the equation as follows:
[tex]\begin{gathered} ln(\frac{1000000}{27000})=ln(1.31209447568)^x=xln(1.31209447568) \\ \\ \frac{ln(\frac{1000000}{27000})}{ln(1.31209447568)}=x \\ \\ x=13.2974595282\text{ hours} \end{gathered}[/tex]Therefore, if the population increases exponentially with time, the number of bacteria will be one million around 13.2975 hours.
Therefore, in summary, we have:
When is the number of bacteria one million if:
a) Does the number increase linearly with time?
It will be 62.3718 hours
b) The number increases exponentially with time?
It will be around 13.2975 hours
Can you please help me out with the a question
Arc XY = 2π • (PX)/ 4
. = 2π • 5/4
. = 6.28 • 5/4= 31.40/4 = 7.85
Then answer is
Option G) 7.854
A cake is cut into 12 equal slices. After 3 days Jake has eaten 5 slices. What is his weekly rate of eating the cake?
5
36
35
36
cakes/week
cakes/week
1
1 cakes/week
35
01. cakes/week
4
Answer:
11.2 Slices / Week
Step-by-step explanation:
We know that Jake has eaten 5 slices of cake in 3 days. You can divide 5 / 3 to get an average of 1.6 slices of cake being eaten per day. The question asks what the weekly rate or eating the cake will be, do you need to multiple 1.6 x 7 for the total amount of cake eaten per week, which is 11.2 slices!
Answer:
11.6
explanation
we have 7 days.
7days-3days =4
in 3 days he has eaten 5 slices
again 4-3 days=1
so in 6 days he has eaten 10 slices
we have 1 day left.so if he eats 5 slices in 3 day,how many he eat slices in 1 day?5/3=1.6
10+1.6=11.6
[tex]((1.25 \times {10}^{ - 15} ) \times (4.15 \times {10}^{25} )) \div ((2.75 \times {10}^{ - 9}) \times (3.4299 \times {10}^{8} ))[/tex]solve. final answer in scientific notation
done
[tex]\text{result = 5.4999 x 10}^{10}[/tex][tex]Inscientificnotation=5.4999x10^0[/tex]Pure acid is to be added to a 10% acid solution to obtain 90L of 81% solution. What amounts of each should be used?How many liters of 100% pure acid should be used to make the solution? 04
Let's use the variable x to represent the amount of pure acid and y to represent the amount of 10% acid.
Since the total amount wanted is 90 L, we can write the equation:
[tex]x+y=90[/tex]Also, the final solution is 81%, so we can write our second equation:
[tex]100\cdot x+10\cdot y=81\cdot(x+y)[/tex]From the first equation, we can solve for y and we will have y = 90 - x.
Using this value in the second equation, we have:
[tex]100x+10(90-x)=81(x+90-x)[/tex]Solving for x, we have:
[tex]\begin{gathered} 100x+900-10x=81\cdot90 \\ 90x+900=7290 \\ 90x=7290-900 \\ 90x=6390 \\ x=\frac{6390}{90} \\ x=71 \end{gathered}[/tex]Therefore the amount of pure acid to be used is 71 L and the amount of 10% acid is 19 L.
A cash register contains only five dollar and ten dollar bills. It contains twice as many fives as tens and the total amount of money in the cash register is 740 dollars. How many tens are in the cash register?
ANSWER
There are 37 tens in the cash register
EXPLANATION
Given that;
The total amount in the cash register is $740
The cash register contain five dollar and ten dollar
Follow the steps below to find the number of ten dollar in the cash register.
Let x represents the number of $5 and $10 in the cash register.
Recall, that the register contain twice as many $5 as ten dollars and this can be expressed mathematically as
[tex]\text{ 5\lparen2x\rparen+ 10\lparen x\rparen= 740}[/tex]Evaluate x in the above expression
[tex]\begin{gathered} \text{ 10x + 10x = 740} \\ \text{ 20x = 740} \\ \text{ Divide both sides by 20} \\ \text{ }\frac{\text{ 20x}}{\text{ 20 }}\text{ = }\frac{\text{ 740}}{\text{ 20}} \\ \text{ x = 37} \end{gathered}[/tex]Therefore, we have 37 tens in the cash register
Calculate the slope of the given line using either the slope formula m = y 2 − y 1 x 2 − x 1 or by counting r i s e r u n . Simplify your answer. You can choose your method.
The slope of the line that passes through points (x1, y1) and (x2, y2) is computed as follows:
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]Replacing with the points (-8, 3) and (0,1) we get:
[tex]m=\frac{1-3}{0-_{}(-8)}=\frac{-2}{8}=-\frac{1}{4}[/tex]On July 31, Oscar Jacobs checked out of the Sandy Beach hotel after spending four nights. The cost of the room was $76.90 per night. He wrote a check to pay for his four-night stay. What is the total amount of his check, expressed in words?
On July 31, Oscar Jacobs checked out of the Sandy Beach hotel after spending four nights. The cost of the room was $76.90 per night. He wrote a check to pay for his four-night stay. What is the total amount of his check, expressed in words?
step 1
Multiply $76.90 by 4
76.90*4=$307.6
so
expressed in words is
three hundred seven and six tenths3. Trapezoid JKLM with vertices J(-4, 3), K(-2, 7),L(2,7), and M(3, 3) in the line y = 1.what would the reflection coordinates be
First, we graph the trapezoid and the line
If we reflect the figure across the line y = 1, then we get the following figure
As you can observe in the graph, the vertices would be J'(-4,-1), K'(-2,-5), L'(2,-5), and M'(3,-1).
farm stand has cherries on 2 shelves. Each shelf has 4 boxes. Each box has 8 ounces of cherries. How many ounces of cherries are displayed in all? Write an expression that represents the amount.
64 ounces of cherries are displayed in all in the farm stand.
According to the question,
We have the following information:
Farm stand has cherries on 2 shelves.
Number of boxes in each shelf = 4 boxes
So, the number of boxes in 2 shelves will be (2*4) or 8.
Ounces of cherries in each box = 8 ounces
Now, the ounces of cherries in 8 boxes can be easily found by multiplying the ounces of cherries in 1 box by the number of total boxes.
Ounces of cherries in 8 boxes = (8*8) ounces
Ounces of cherries in 8 boxes = 64 ounces
Now, the expression that represents the amount is (number of shelves*number of boxes*ounces of cherries in each box).
Hence, the ounces of cherries displayed in all is 64 ounces.
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A number from 1-40 is chosen at random. Find each probability.1. Pleven | at least 12)2. P(perfect square | odd)3. P(less than 25 | prime)4. P(multiple of 3 | greater than 15)
Given:
Numbers from 1 - 40
Let's find the probability of:
Pleven | at least 12)
Where:
Even numbers = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40 = 20 numbers
Even numbers that are at least 12 = 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40 = 15 numbers.
Numbers that are at least 12 = 29 numbers
Therefore, to find the probability, we have:
[tex]P(even|atleast12)=\frac{P(even\text{ and at least 12\rparen}}{P(at\text{ least 12\rparen}}[/tex]Where:
[tex]\begin{gathered} P(even\text{ and at least 12\rparen = }\frac{15}{40}=0.375 \\ \\ P(at\text{ least 12\rparen= }\frac{29}{40}=0.725 \end{gathered}[/tex]Therefore, we have:
[tex]\begin{gathered} P(even|atleast12)=\frac{0.375}{0.725} \\ \\ P(even|atleast12)=0.52 \end{gathered}[/tex]Therefore, the probability that a number chosen at random is even given that it is at least 12 is 0.52
ANSWER:
0.52
QuestionFind the equation of a line that contains the points (-6, 3) and (5,-8). Write the equation in slope-intercept form.
ANSWER
y = -x - 3
STEP BY STEP EXPLANATION
Step 1: The given points are:
(-6, 3) and (5, -8)
Step 2: The slope-intercept form is
[tex]y\text{ = mx + c}[/tex]where m is the slope and c is the intercept
Step 3: Find the slope m
[tex]\begin{gathered} \text{slope (m) = }\frac{y_2-y_1}{x_2-x_1} \\ \text{m = }\frac{-8_{}-\text{ 3}}{5\text{ - (-6)}} \\ m\text{ = }\frac{-11}{11}\text{ = -1} \end{gathered}[/tex]Step 4: Solve for intercept c using either of the points
[tex]\begin{gathered} y\text{ = mx + c} \\ c\text{ = y - mx} \\ c\text{ = 3 - (-1)(-6)} \\ c\text{ = 3 - 6} \\ c\text{ = -3} \end{gathered}[/tex]Step 5: Re-writing the slope-intercept form to include the values of m and c
[tex]\begin{gathered} y\text{ = mx + c} \\ y\text{ = -x - 3} \end{gathered}[/tex]Hence, the equation of the line in slope-intercept form is y = -x - 3
If the area of a rectangular field is x2 – 3x + 4 units and the width is 2x – 3, then find the length of the rectangular field.x2- 3 x + 42 x − 3 unitsx2 - 3x + 4 units2x - 3 units3x + 4 units
Solution
We are given the following
[tex]\begin{gathered} Area=x^2-3x+4 \\ \\ Width=2x-3 \\ \\ Length=? \end{gathered}[/tex]Using the Area of a Rectangle we have
[tex]\begin{gathered} Area=lw \\ \\ l=\frac{A}{w} \\ \\ l=\frac{x^2-3x+4}{2x-3} \end{gathered}[/tex]Therefore, the answer is
[tex]\frac{x^{2}-3x+4}{2x-3}units[/tex]I just need to know if You just have to tell me if the circles are open or closed.
Solution
- The solution is given below:
[tex]\begin{gathered} y-2<-5 \\ y-2>5 \\ \\ \text{ Add 2 to both sides} \\ \\ y<-5+2 \\ y<-3 \\ \\ y-2>5 \\ y>5+2 \\ y>7 \end{gathered}[/tex]- Thus, we have:
[tex]\begin{gathered} y<-3 \\ or \\ y>7 \end{gathered}[/tex]- Thus, the plot is:
Identify whether the following real world examples should be modeled by a linear quadratic or exponential function
Solution
- Linear:
The general form of a linear function is
[tex]\begin{gathered} y=ax+b \\ where, \\ a,\text{ and b are constants} \end{gathered}[/tex]- Quadratic:
The general form of a quadratic function is:
[tex]\begin{gathered} y=ax^2+bx+c \\ where, \\ a,b,c\text{ are constants} \end{gathered}[/tex]- Exponential:
The general form of an exponential function is:
[tex]\begin{gathered} y=ab^x \\ where, \\ a,b\text{ are constants} \end{gathered}[/tex]- Now that we know the general forms of these functions, we can proceed to solve the question.
- The amount a person is paid per hour in wages is the amount that the person collects for every hour that he works
- Let us imagine that a person receives $a for every hour worked.
- This means that:
After 1 hour, the person makes $a
After 2 hours, the person makes $a + $a = $2a
After 3 hours, the person makes $a + $a +$a = $3a
- We can therefore generalize as follows:
Thus, after x hours, the person makes:
[tex]x\times a=\$ax[/tex]- Thus, the function representing the amount a person makes per hour of work is given by:
[tex]y=ax[/tex]- Comparing this result with the 3 function definitions above, we can see that this corresponds to a Linear function
Final Answer
The answer is Linear
The height, in feet, of a particle from the ground is given by the function s(t) = 1.512 + 20r, where 0 ≤ ≤ 17.
Find the velocity of the particle at t = 4.
Answer
feet per second
The velocity is v= 30.6 ft/ sec.
What is a velocity?Velocity defines the direction of the movement of the body or the object. Speed is primarily a scalar quantity. Velocity is essentially a vector quantity. It is the rate of change of distance. It is the rate of change of displacement.
Given that,
We have given the height
s(t) = 0.2[tex]t^{3}[/tex] + 21t, where 0 ≤ [tex]x[/tex] ≤ 17.
To find the velocity we have to differentiate s(t) wrt to t.
s(t) = 0.2[tex]t^{3}[/tex] + 21t
= 0.6[tex]t^{2}[/tex]+21
velocity of the particle at t = 4
s(4) = 0.6*[tex]4^{2}[/tex]+21
= 9.6+21
= 30.6
v= 30.6 ft/ sec
Hence, The velocity is v= 30.6 ft/ sec.
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solve for x in the parallelogram below
Solve: 5|4x+5|−2≤33 Give your answer as an interval. If no solutions exists - enter No solutions.
The expression given is,
[tex]5|x-3|+3>7[/tex]Subtract 3 from both sides
[tex]\begin{gathered} 5|x-3|+3-3>7-3 \\ 5|x-3|>4 \end{gathered}[/tex]Divide both sides by 5
[tex]\begin{gathered} \frac{5|x-3|}{5}>\frac{4}{5} \\ |x-3|>\frac{4}{5} \end{gathered}[/tex]Apply absolute rule:
[tex]\begin{gathered} x-3<-\frac{4}{5}\text{ or x-3>}\frac{4}{5} \\ \end{gathered}[/tex]Add 3 to both sides
[tex]\begin{gathered} x-3+3<-\frac{4}{5}+3\text{ or x-3+3>}\frac{4}{5}+3 \\ x<\frac{11}{5}\text{ or x>}\frac{19}{5} \end{gathered}[/tex]Therefore, the answer has the form:
[tex](-\infty,A)\cup(B,\infty)[/tex]Hence, the solution using interval notation is
[tex](-\infty,\frac{11}{5})\cup(\frac{19}{5},\infty)[/tex]what is 9/36 simplified?
Answer:
1/4
Step-by-step explanation:
it can be simplified by dividing both the numerator and denominator with 9.
[tex] \displaystyle \large{ \sf{ \frac{9}{36}}} [/tex]
[tex]\displaystyle \large{ \sf{ \frac{9}{36} = \frac{ \cancel9}{ \cancel3 \cancel6} }}[/tex]
[tex]\displaystyle \large{ \bf{ = \frac{1}{4} }}[/tex]
simplest form is 1/4
The circle has center O. Its radius is 4 cm, and the central angle a measures 30°. What is the area of the shaded region?Give the exact answer in terms of pi, and be sure to include the correct unit in your answer
Explanation
The area of a portion of a circle with radius 'r' and central angle 'a' in radians is:
[tex]A_{\text{portion}}=\frac{1}{2}\cdot r^2\cdot a[/tex]In this problem, the radius is r = 4cm, and the angle a = 30º.
First we have to express the angle in radians:
[tex]a=30º\cdot\frac{\pi}{180º}=\frac{1}{6}\pi[/tex]And now we can find the area of the shaded region:
[tex]\begin{gathered} A=\frac{1}{2}\cdot(4\operatorname{cm})^2\cdot\frac{1}{6}\pi \\ A=\frac{1}{2}\cdot16\operatorname{cm}^{2}\cdot\frac{1}{6}\pi=\frac{4}{3}\pi \end{gathered}[/tex]Answer
The area of the shaded region is:
[tex]A=\frac{4}{3}\pi cm^{2}[/tex]Janelle says that lines l and m are skew lines. Planes B and A intersect. Plane B is vertical and contains vertical line n. Plane A is horizontal and contains horizontal line m. Line m and n are perpendicular. Line l is on plane A and it is slightly diagonal. Is Janelle correct? Yes, because the lines are not parallel. Yes, because the lines will intersect. No, because the lines are in the same plane. No, because the lines are perpendicular.
Lines are parallel and on the same plane. The final answer, "No, because the lines are in the same plane," is the proper response.
What is a line that is perpendicular?
Perpendicular lines are those that cross at a perfect right angle. Parallel lines are those that are always the same distance apart from one another.
The question is incomplete.
Please see the accompanying image for a comprehensive explanation of the question.
Line l and line m are the two lines that are depicted in the image.
The skew lines are in different planes and do not overlap, as far as we are aware.
Therefore,
Lines are parallel and on the same plane. The final answer, "No, because the lines are in the same plane," is the proper response.
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Deter mine the intervals for which the function shown below is increasing
Answer:
The interval at which the function is increasing is from x = -2 to x = 0. In interval notation, it is (-2, 0).
Explanation:
See the graph below for the pattern of the function.
As you can see above, from x = -∞ until x = -2, the value of the function decreases from y = +∞ to y = -7.
Then, starting at x = -2 to x = 0, the value of the function increases from y = -7 to y = -3.
Lastly, starting at x = 0 to +∞, the value of the function decreases again from y = -3 to -∞.
Hence, the interval at which the function is increasing is at (-2, 0).
I have a practice problem in the calculus subject, I’m having trouble solving it properly
The limit of a function is the value that a function approaches as that function's inputs get closer and closer to some number.
The question asks us to estimate from the table:
[tex]\lim _{x\to-2}g(x)[/tex]To find the limit of g(x) as x tends to -2, we need to check the trend of the function as we head towards -2 from both negative and positive infinity.
From negative infinity, the closest value we can get to before -2 is -2.001 according to the values given in the table. The value of g(x) from the table is:
[tex]\lim _{x\to-2^+}g(x)=8.02[/tex]From positive infinity, the closest value we can get to before -2 is -1.999 according to the values given in the table. The value of g(x) from the table is:
[tex]\lim _{x\to-2^-}g(x)=8.03[/tex]From the options, the closest estimate for the limit is 8.03.
The correct option is the SECOND OPTION.
which three statements are true about the line segment CBit's the radius of the circleit is the circumference of the circleit is a cordit is 6cm longit is diameter of the circle it is 7cm longit is 1.75cm long
Answer:
It is the diameter of the circle
it is 7 cm
it is a chord
Explanation:
First, we notice that the line segment CB passes through the centre of the circle and its endpoints touch the circumference - this tells us that CB is the diameter.
Furthermore, any line segment whose endpoints lie on the circumference of the circle is a chord (meaning that the diameter is the longest chord), and so we deduce that CB is also a chord.
Since CB is the diamter, its length is 2 times the radius. The raduis of the circle we know is DA = 3.5 cm; therefore, the dimater is CB = 2 DA = 2 * 3.5 = 7 cm.
Hence, the correct choices are:
It is the diameter of the circle
it is 7 cm
it is a chord
find the slope. A. y= -1/2x - 19/2.
The equation of the line follows the following general structure:
[tex]y=mx+b[/tex]Where m is the slope of the line and b is the y intercept.
Find the corresponding values in the given formula, this way:
In the given equation, m has a value of -1/2, it means the slope is -1/2.
The distance around a water fountian is 150 inches what is the distance from the edge of the fountian to the center
Answer:
The distance from the edge of the fountain to the centre is approximately 23.87 inches.
The water fountain forms a circle. The distance around the water fountain is the circumference of the circle formed.
Therefore,
circumference = 2πr
150 = 2πr
The distance from the edge of the fountain to the centre is the radius of the circle formed. Therefore,
75 = πr
r = 75 / 3.14159
r = 23.8732616287
r = 23.87 inches
The distance from the edge of the fountain to the centre is approximately 23.87 inches.
I got stuck and I need help on this I would appreciate the help:0
1) In this problem, we can see that this is an isosceles right triangle.
2) So, one way of solving it is to make use of the Pythagorean theorem. Note that an isosceles triangle has two congruent sides, so we can write out:
[tex]\begin{gathered} a^2=b^2+c^2 \\ b=c \\ 9^2=x^2+x^2 \\ 81=2x^2 \\ 2x^2=81 \\ \frac{2x^2}{2}=\frac{81}{2} \\ x^2=\frac{81}{2} \\ \sqrt[]{x^2}=\sqrt[]{\frac{81}{2}} \\ x=\frac{9}{\sqrt[]{2}} \end{gathered}[/tex]Usually, we rationalize it. But since the question requests the denominator to be a rational one, so this is the answer.
Last year, Kevin had $10,000 to invest. he invested some of it in an account that paid 6% simple interest per year, and he invested the rest in an account that paid 10% simple interest per year. after one year, he received a total of $920 in interest. how much did he invest in each account?first account:second account:
Simple interest is represented by the following expression:
[tex]\begin{gathered} I=\text{Prt} \\ \text{where,} \\ I=\text{ interest} \\ P=\text{principal} \\ r=\text{interest rate in decimal form} \\ t=\text{ time (years)} \end{gathered}[/tex]We need to create a system of equations:
Let x be the money invested in the account that paid 6%
Let y be the money invested in the account that paid 10%
So, he received a total of $920 in interest, then:
[tex]920=0.06x+0.1y\text{ (1)}[/tex]And we know that money invested must add together $10,000:
[tex]x+y=10,000\text{ (2)}[/tex]Then, we can isolate y in equation (2):
[tex]y=10,000-x[/tex]Now, let's substitute y=10,000-x in the equation (1):
[tex]\begin{gathered} 920=0.06x+0.1(10,000-x) \\ 920=0.06x+1000-0.1x \\ 0.1x-0.06x=1,000-920 \\ 0.04x=80 \\ x=\frac{80}{0.04} \\ x=2,000 \end{gathered}[/tex]That means, he invested $2,000 in the account that paid 6% simple interest. Now, having x, we are going to substitute x in the second equation (2):
[tex]\begin{gathered} y=10,000-x \\ y=10,000-2,000 \\ y=8,000 \end{gathered}[/tex]He invested $8,000 in the account that paid 10% simple interest per year.
I inserted a picture of the question please state whether the answer is a b c or d PLEASE GIVE A VERY VERY SHORT EXPLANATION
The 30-60-90 triangle is given by
AS we can note , the hypotenuse is twice as long as the shorter leg. Additionally, the longer leg is square root of 3 tines as long as the shorter leg. Therefore, the answer is option C and F
Use the substitution u = (2x - 2) to evaluate the integral x³e(^2x^4-2) dx
The substitution u = (2x - 2) to the integral x³e(^2x^4-2) dx is (2x – 2)/4 +c
What is meant by integral?In mathematics, an integral assigns numerical values to functions in order to describe concepts like displacement, area, volume, and other outcomes of the combination of infinitesimally small data. Integral discovery is a process that is referred to as integration. One of the fundamental, crucial operations of calculus, along with differentiation, is integration[a]. It can be used to solve issues in mathematics and physics involving, among other things, the volume of a solid, the length of a curve, and the area of an arbitrary shape. The integrals listed here are those that fall under the category of definite integrals, which can be thought of as the signed area of the region in the plane that is enclosed by the graph of a particular function between two points on the real line.Therefore,
Use the substitution
U = (2x -2)
to evaluate integral x³e(^2x^4-2) dx
let u = 2x -2
du = x dx or dx =du/2
u = 2x-2
du = d(2x – 2)
du = 2dx
dx = du/2
∫ (2x -2)dx = ∫u du/2
=1/2 ∫u du
= ½ u square /2 +c
= u square /4 +c
= (2x – 2)/4 +c
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Simplify cot(t)/csc(t)-sin(t) to a single trig function
The single trig function that simplifies the function is sec(t)
How can we simplify the function?Trigonometry deals with the functions of angles and how they're applied.
Given cot(t)/csc(t)-sin(t)
since csc(t) = 1/sin(t) , we have:
[tex]\frac{ cot(t)}{csc(t)-sin(t)} = \frac{cot(t)}{\frac{1}{sin(t)} - sin(t) }[/tex]
[tex]\frac{ cot(t)}{csc(t)-sin(t)} = \frac{cot(t)}{\frac{1-sin^{2}(t) }{sin(t)} }[/tex]
since:
cos²(t) = 1 - sin²(t)
Therefore we have:
cot(t) / csc(t)-sin(t) = cot(t)/ cos²(t)/sin(t)
cot(t) / csc(t)-sin(t) = cot(t) / cos(t).cos(t)/sin(t)
Since cos(t) / sin(t) = 1/tan(t) = cot(t)
Therefore:
cot(t) / csc(t)-sin(t) = cot(t)/ cot(t)×cos(t)
cot(t) / csc(t)-sin(t) = 1/cos(t)
Since 1/cost = sec(t)
Finally, cot(t) / csc(t)-sin(t) is sec(t).
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