1) The differential equation is: dP/dt=kP
2) P(t)=8000[tex]e^{-0.28t}[/tex]
3) The solution at the specified value of the independent variable is 1490.991
What is meant by differential equation?In practical contexts, derivatives frequently represent the rates of change of physical quantities, and the differential equation establishes a relationship between the two. As a result of the frequent occurrence of these correlations, differential equations are widely employed in a wide range of disciplines, including engineering, physics, economics, and biology.
Only the simplest differential equations can be resolved explicitly, but many aspects of a given differential equation's solutions can be determined without doing an exact calculation.
1) The differential equation is: dP/dt=kP
Here k is the proportionality constant
2) Now, we have to solve the differential equation on dP/dt=kP
∫dP/P=∫kdt+lnP₀
lnP₀=kt+lnP₀
lnP/P₀=kt
P=P₀[tex]e^{kt}[/tex]
Here P₀ is the constant of integration.
P(t)=P₀[tex]e^{kt}[/tex]
And they have given that,
P(0)=8000=P₀e⁰
P₀=8000
P(1)=5200
5200=8000[tex]e^{k}[/tex]
k=-2log(2)+log(3)≈ -0.28
Therefore, the required function is:
P(t)=8000[tex]e^{-0.28t}[/tex]
3) Now, we have to put t=6 in the above equation:
P(6)=8000[tex]e^{-0.28*6}[/tex]
P(6)=1490.991
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In class of students, the following data table summarizes how many students play and instrument or a sport. What is the probability that a student who plays a sport also plays an instrument?
A student selected at random from the class has a 0.5 is 50% chance of not participating in sports, according to the probability notion.
What is the probability?A probability is calculated by dividing the entire number of possible outcomes by the desired outcomes.
In this issue:
The total number of students is 30 (8 + 7 + 3 + 12).
3 + 12 = 15 of the total don't participate in sports.
Hence:
p =D/T = 15/30 = 0.5
There is a 50% chance that a student who was randomly selected from the class does not participate in sports.
A student selected at random from the class has a 0.5 is 50% chance of not participating in sports, according to the probability notion.
The complete question is :in a class of students, the following data table summarizes how many students playing instrument or a sport. What is the probability that student plays an instrument given that they play a sport?
play an instrument and a sport: 3
play an instrument but does not play a sport: 8
does not play an instrument but plays a sport: 7
does not play an instrument or a sport: 12
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a rectangle is 19 feet long and 3 feet wide find the area
The area of a rectangle is given as:
Area = Length x width
The area of the rectangle is 57 square feet.
What is a rectangle?A rectangle is a two-dimensional shape where the length and width are different.
The area of a rectangle is given as:
Area = Length x width
We have,
Length = 19 feet
Width = 3 feet
The area of the rectangle.
= 19 x 3
= 57 square feet.
Thus,
The area of the rectangle is 57 square feet.
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BRAINLEST
PLEASE HELP
Graph f(x)=|x−6|−4.
Use the ray tool to graph the function.
The graph of the given function f(x)=|x−6|−4 is shown where the line is parallel to the x-axis.
What are graphs?A graph is a structure that resembles a set of items in discrete mathematics, more specifically in graph theory, in which some pairs of objects are conceptually "connected."
The items are represented by mathematical abstractions known as vertices, and each pair of connected vertices is referred to as an edge.
Write the equation in the form y = MX + b to determine the slope m and the y-intercept.
This will allow you to graph the equation using the slope and y-intercept (0, b).
So, the given function is: f(x)=|x−6|−4
Take f(x) as 0 which means:
f(x)=|x−6|−4 = 0
Now, graph the function as follows:
(Refer to the graph attached below)
The line is parallel to the x-axis.
Therefore, the graph of the given function f(x)=|x−6|−4 is shown where the line is parallel to the x-axis.
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CAN SOMEONE HELP WITH THIS QUESTION?✨
The required values of a and b for the given case are 10000 and 15.08 respectively.
How to find the decay rate of an exponential function?An exponentially decaying function has a general form f(x) = (1 -r)ˣ, where r is called the decay rate. it can be found by having any two values of x for which f(x) is known.
The given problem can be solved as follows,
The function is f(x) = abˣ
And, the given points on the function are (0, 10000) and (3, 3430).
Substitute x = 0 and f(x) = 10000 in the f(x) to obtain,
f(0) = ab⁰
=> 10000 = a
Again, substitute x = 3 and f(x) = 3430 in the f(x) to obtain,
f(3) = ab³
=> 3430 = 10000 × b³
=> b = ∛0.3430
=> b = 15.08
Hence, the values of a and b for the given exponential function are 10000 and 15.08 respectively.
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Find the coordinates of a point on a circle with radius 10 corresponding to an angle of
325∘
To find the coordinates of a point on a circle with radius 10 corresponding to an angle of 325 degrees, you can use the following formula:
(x, y) = (r * cos(θ), r * sin(θ))
where (x, y) are the coordinates of the point, r is the radius of the circle, and θ is the angle in radians.
To convert the angle from degrees to radians, you can use the following formula:
θ (radians) = θ (degrees) * (π / 180)
Substituting the values into the formula and converting the angle to radians, you get:
(x, y) = (10 * cos(325 * (π / 180)), 10 * sin(325 * (π / 180)))
= (-5.5, 9.5)
Therefore, the coordinates of the point on the circle with radius 10 corresponding to an angle of 325 degrees are (-5.5, 9.5).
What are the coordinates point?
The coordinates of a point in a coordinate system are a set of numerical values that specify the position of the point in relation to an origin. In a two-dimensional coordinate system, such as the Cartesian coordinate system, a point is represented by an ordered pair (x, y), where x and y are the coordinates of the point. The x-coordinate represents the point's position along the horizontal axis, and the y-coordinate represents the point's position along the vertical axis.
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Calculate the dosage (in mg/min) to the nearest tenth using the calculations method, ratio and proportion or dimensional analysis, that you choose. (Simplify your answers completely.)
An infusion of 4 g in 500 mL is ordered at 60 mL/hr.
The required value of dosage in mg/min is given as 8 mg/min.
What is the application ratio and proportion?A ratio is the relation between two numbers as a / b. A proportion is the equality of two ratios as a / b = c / d.
Ratio and proportion can be applied to solve Mathematical problems dealing with unit values of the quantities.
Given that,
The dose administered per hour is 60 mL/hr.
And, the infusion is given as 4g for 500 mL.
In order to calculate the dosage in mg/min, ratio and proportion method can be used as follows,
The 500 mL is for 4g.
Then for 1 mL, there is 4/500 g.
Thus, for 60 mL, there is 4/500 × 60 g.
Since 1 g = 1000 mg.
The above expression can be written as,
4/500 × 60 × 1000 mg = 480 mg
Now, the dosage can be calculated as,
480 mg/hr
= 480/60 mg/min
= 8 mg/min
Hence, the dosage in milligram per minutes is given as 8 mg/min.
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Use the results from a survey of a simple random sample of 1055 adults. Among the 1055 respondents, 77% rated themselves as above average drivers. We want to test the claim that 7/10 of adults rate themselves as above average drivers. Complete parts (a) through (c).
The p-value of the test is 0, which is less than the standard significance level of 0.05, thus we can conclude that the proportion respondents who rated themselves as above average drivers is greater than 0.7.
What is simple random sample?We may test the following using the example data:
In reality, 812.35 respondents identified as better drivers than average.
We can infer that the proportion of respondents who rated themselves as above average drivers is greater than 0.7 because the test's p-value is 0, which is less than the standard significance level of 0.05. 77% of the sample of 1055 adults rated themselves as above average drivers, so 0.77(1055) = 812.35.
As a result, 812.35 respondents actually assessed themselves as better drivers than average.
We examine if the proportion is 7/10=0.7 at the null hypothesis.
H0 is that, with p = 0.7.
If the proportion is larger than 70%, we test for the alternative hypothesis, which is H1:p>=0.7.
The test statistic comes from:
The parameters for this issue are p dash = 0.77, p = 0.7, and n = 1055.
As a result, the test statistic's value is:
z = 351.758 root
The likelihood of discovering a sample proportion above 0.77, calculated as 1 minus the p-value of z = root of 351.758, is the test's p-value.
According to the z-table, the p-value for z = root of 351.758 is 1, and 1 - 1 equals 0.
Since the test's p-value is zero and is below the conventional significance level of 0.05, we can infer that a higher proportion of respondents than 0.7 evaluated themselves as above-average drivers.
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Music stalls Games stalls Handicraft stalls Ks 3000 Food stalls Ks 4800 (a) How much money was collected by the music stalls? (b) What fraction of the total amount of money was collected by the game stalls? (c) What percentage of the total amount of money was collected by the food stalls? (d) What was the ratio of the money collected by the handicraft stalls to the money collected by the music stalls?
Answer:
Step-by-step explanation:
a. The music stalls collected Ks 3000.
b. The game stalls collected Ks 4800, which is a fraction of the total amount of money collected. To find the fraction, we can divide the amount collected by the game stalls by the total amount collected by all the stalls:
$\frac{Ks 4800}{Ks 3000 + Ks 4800 + Ks 3000} = \frac{Ks 4800}{Ks 10800} = \frac{2}{9}$
Therefore, the game stalls collected $\frac{2}{9}$ of the total amount of money.
c. The food stalls collected Ks 4800, which is a percentage of the total amount of money collected. To find the percentage, we can divide the amount collected by the food stalls by the total amount collected by all the stalls and multiply by 100%:
$\frac{Ks 4800}{Ks 3000 + Ks 4800 + Ks 3000} \times 100% = \frac{Ks 4800}{Ks 10800} \times 100% = 44.44%$
Therefore, the food stalls collected 44.44% of the total amount of money.
d. The handicraft stalls collected Ks 3000, and the music stalls collected Ks 3000. The ratio of the money collected by the handicraft stalls to the money collected by the music stalls is therefore 1:1. We can write this ratio as $\frac{1}{1} = 1$, or simply as 1.
A friend lends you $520, which you agree to repay with 8% interest. How much will you have to repay?
Answer:
Total interest paid:$112.65
Total payments:$632.65
Step-by-step explanation:
Please help! I have little time to do this
$2, 900 is invested in an account earning 6.1% interest (APR), compounded quarterly. Write a function showing the value of the account after t years, where the annual growth rate can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage of growth per year (APY), to the nearest hundredth of a percent.
A function showing the value of the account after t years, where the annual growth rate can be found from a constant in the function is f(t) = 2900 × (1.0624)^t and APY is 6.24%.
What is Compound Interest?Compound interest in simple words is the interest earning for your savings with the interest which those savings earn.
The formula for calculating the amount in Compound Interest is,
A = P( 1 + [tex]\frac{r}{n}[/tex] )ⁿᵇ
where A is the amount, P is the principal, r is the rate of interest, n is the number of times interest compounded in a year and b is the number of years.
Here, P = $2900, r = 6.1% = 0.061, n = 4 (quarterly) and b = t years
A = 2900 (1 + [tex]\frac{0.061}{4}[/tex])^4t
A = 2900 (1.0153)^4t
A = 2900 × (1.0624)^t
So A can be said as a function of t.
f(t) = 2900 × (1.0624)^t
Percentage of growth per year (APY) = ( 1 + [tex]\frac{r}{n}[/tex] )ⁿ - 1
= (1 + [tex]\frac{0.061}{4}[/tex])⁴ - 1
= 1.0624 - 1
= 0.0624 = 6.24%
Hence the function of the value of the account after t years is f(t) = 2900 × (1.0624)^t, where the value 1.0624 is a constant.
Percentage of growth per year is 6.24%.
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what is 2 x 2 x 2 using an exponent, in cubic units , then evaluate
Answer:
2^3 = 8
Step-by-step explanation:
Consider a circle whose equation is x2 + y2 – 2x – 8 = 0. Which statements are true? Select three options. The radius of the circle is 3 units. The center of the circle lies on the x-axis. The center of the circle lies on the y-axis. The standard form of the equation is (x – 1)² + y² = 3. The radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
The center of the circle lies on the x-axis. The radius of the circle is 3 units. The radius of this circle is the same as the radius of the circle whose equation is x² + y² is 9.
What is radius of a circle?The radius of a circle is the distance between the circle's centre and any point on its circumference. It is usually represented by the letters 'R' or 'r'. This number is significant in almost all circle-related formulas. A circle's area and circumference are also measured in terms of radius.
The standard equation of a circle is expressed as:
x² + y²+2gx+2fy+C=0
Centre is (-g, -f)
radius = √g²+f²-C
Given a circle whose equation is 2+ y²-2x-8=0.
Get the centre of the circle
2gx = -2x
2g=-2
g=-1
Similarly, 2fy = 0
f=0
Centre = (-(-1), 0) = (1, 0)
This shows that the center of the circle lies on the x-axis
r = radius = √g²+f²-C
radius = √1²+02-(-8)
radius =√9 = 3 units
The radius of the circle is 3 units.
For the circle x² + y² = 9, the radius is expressed as:
r² = 9
r = 3 units
Hence the radius of this circle is the same as the radius of the circle whose equation is x² + y² = 9.
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I cannot figure out this problem
a) The Test statistic of the hypothesis is; 0.81
b) The p-value from the test statistic is; 0.20897.
c) The final conclusion is that there is not sufficient evidence to reject the null hypothesis that (p1 - p2) = 0
How to find the test statistic?Test statistic is defined as a number, calculated from a statistical test, used to find if your data could have occurred under the null hypothesis. In this case, we are dealing with proportions and as such the test statistic has the formula;
z = (p1 - p2)/standard error
p1 = 53/90 = 0.59; s1 = √(0.59 * 0.41/90) = 0.052
p2 = 48/90 = 0.53; s2 = √(0.53 * 0.47/90) = 0.053
Hence, for the distribution of differences, the mean and the standard error are given as follows:
Mean: p = 0.59 - 0.53 = 0.06
Standard error: s = √(0.052² + 0.053²) = 0.074
a) Test statistic = p/s = 0.06/0.074
Test statistic = 0.81
b) From p-value from z-score calculator, we get that;
p-value = 0.20897.
c) The p-value is greater than the significance value of 0.05, we fail to reject the null hypothesis and conclude that there is not suffucuent evidence to reject the null hypothesis that (p1 - p2) = 0
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what is the first significant figure in the number 0.00892?
Answer:
0.009
Step-by-step explanation:
0.00892
first significant figure will be
0.00900
or = 0.009
i hope this helps
Answer:
0.009
Step-by-step explanation:
0.00892(a significant figure is a number greater than zero)
so the first number after zero is 8
the next number is 9, so round 9 up and it becomes 1.
Add 1 to 8 to give you 9
=0.00900 or 0.009
Hope it helped you
convert the point (4,197) from polar coordinates into rectangular coordinates. round your answers to the nearest tenth (1 decimal place).
The rectangular coordinates are (-3.825,-1.169).
In a rectangular coordinate system, we were plotting points based on an ordered pair of (x, y). In the polar coordinate system, the ordered pair will now be (r, θ). The ordered pair specifies a point’s location based on the value of r and the angle, θ, from the polar axis.
The value of r can be positive, negative, or zero. The sign of r is very important in locating the exact position of the point. The absolute value of r, |r|, is the distance between the point and the pole.
1. If r is positive (r > 0) then the point lies on the terminal side of θ
2. If r is negative (r < 0) then the point lies on the ray opposite of the terminal side of θ
3. If r is zero (r = 0) then the point lies at the pole regardless of θ
Given,
P= (4,197)
Finding rectangular coordinates (x,y)
x = r cosθ
x = 4 cos 197
x = 4* (-0.956)
x = -3.825
y = r sinθ
y = 4 sin 197
y = 4*(-0.292)
y = -1.169
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What is the value of x?
Please help me
Answer:
x = 35
Important things to know:
1. A + (x + 110) will equal 180 degrees
2. The interior angles of a triangle will equal 180 degrees.
With these two pieces of knowledge, you can substitute the value of A as "180 - (x+110)" because A = 180-(x+110)
Step-by-step explanation:
Find the point of intersection for the pair of linear equations.
x+y= -8.5
y = 2x + 4.4
A. (-4.3, -4.2)
B. (-4.1, 4.4)
C. (1.9, -3.7)
D. (-3.1, 2.4)
Solve for inequality
Seven more than the quotient of a number $b$ and 45 vis greater than 5 .
Seven more than the quotient of a number b and 45 is greater than 5 then the inequality is b > -90.
What is inequality?
Equations are not always balanced on both sides using a "equal to" symbol in mathematics. Sometimes it can be about a "not equal to" relationship, meaning that something is superior to or inferior to another. In mathematics, an inequality is a relationship between two numbers or other mathematical expressions that results in an unequal comparison.7 more than the quotient of a number b and 45 is greater than 5.
interpretation
7 + b/45 > 5
collect the like terms
b/45>5-7
b/45>-2
cross multiply
∴ b>-2×45
b > -90
Hence, Seven more than the quotient of a number b and 45 is greater than 5 then the inequality is b > -90.
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A piece of rope is 24 1/2 feet long. How many 1/4- foot sections can be cut from it?
NEED HELP ASAP!!
=============================================
Explanation:
1/4 = 0.25
1/2 = 0.5
24 & 1/2 = 24.5
The rope is 24.5 ft long and you want sections of length 0.25 ft each.
Let x be the number of sections we can make.
1 section is 0.25 ft long
x of them combine to 0.25x feet long
Set this equal to 24.5 and solve for x
0.25x = 24.5
x = (24.5)/(0.25)
x = 98
There will be 98 sections of equal length (each being 0.25 ft)
------------------
Another way to look at it:
1 foot of rope gets you four sections of 0.25 ft each
24 feet of rope gets you 24*4 = 96 sections.
Another 0.5 ft adds another 0.5*4 = 2 sections to get to a total of 96+2 = 98
4. A desk is 1 ½ feet wide and 5 feet long. It is
2½ feet above the floor. What is the volume
of space under the desk?
Answer:
gg
Step-by-step explanation:
An election ballot asks voters to select two city commissioners from a
group of eight candidates. In how many ways can this be done?
Answer:
The answer is 4
Step-by-step explanation:
The group candidate is 8
with a division between the number of candidates and the number of selection will the result.
8 candidates ÷ 2 candidates = 4
The way in which 2 candidates can be selected from the pool of 8 candidates to be a city commissioner is 4.
Romney bought a car with $6000 down payment, and paid the remaining cost by installments, which totaled $15000. Romney then sold the car after 10 years for $5500. How much did driving this car cost Romney per year?
Hence, the driving second car cost Romney per year pays is $ [tex]16364[/tex].
What is the selling price?
The selling price of a product or service is the seller’s final price, i.e., how much the customer pays for something. The exchange can be for a product or service in a certain quantity, weight, or measure.
Here given that,
Romney bought a car with $[tex]6000[/tex] down payment, and paid the remaining cost by installments, which totaled $[tex]15000[/tex]. Romney then sold the car after [tex]10[/tex] years for $[tex]5500[/tex].
So, he sold his car in $[tex]15000(6000)=90000000[/tex].
And he is paying installment in every month which is $ [tex]15000[/tex].
So, in case of second car he paied $ [tex]5500[/tex] and his installments of every month would be
[tex]\frac{90000000}{5500}=16364[/tex] $
Hence, the driving second car cost Romney per year pays is $ [tex]16364[/tex].
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Which of the following statements is true of the function g(x)= 1/ x+3 - 5
Please help as soon as possible, and please make sure you say A, B,C, or D at least if not typing which choice. Thanks!
g(x) can be graphed by translating the basic rational function ƒ(x) 1∕x left by 3 units and downward by 5 units.
B)
g(x) can be graphed by translating the basic rational function ƒ(x) 1∕x right by 3 units and downward by 5 units.
C)
g(x) can be graphed by translating the basic rational function ƒ(x) 1∕x right by 3 units and downward by 5 units.
D)
g(x) can be graphed by translating the basic rational function ƒ(x) 1∕x left by 5 units and downward by 3 units.
Answer: none of these
Step-by-step explanation:
okay, your answers b and c are the same so maybe you typed or copied them in wrong
but if the function is g(x)=1/(x+3)-5 than the answer would be…
translate it 3 right and up 5
I hope this helps!
Find the amount A in an account (to the nearest cent) after 10 years if
dA
dt = rA, A(0) = 15,000 and A(8) = $22,377.37
The amount A in an account to the nearest cent, the amount A in the account after 10 years is $30,441.65.
What is the amount A in an account?Generally, To find the amount A in the account after 10 years, we can use the formula for continuous compound interest:
A(t) = A(0) * e^(rt)
where A(t) is the amount in the account after t years, A(0) is the initial amount in the account, r is the interest rate, and t is the number of years.
In this case, we know that A(0) = $15,000, A(8) = $22,377.37, and t = 10 years. We can rearrange the formula to solve for r:
r = (1/t) * ln(A(t)/A(0))
Substituting the known values, we get:
r = (1/10) * ln(22377.37/15000)
Solving this equation, we get r = 0.0658082.
Now that we have the value of r, we can use the formula for continuous compound interest to find the amount A in the account after 10 years:
A(10) = 15000 * e^(0.0658082 * 10)
Solving this equation, we get A(10) = $30,441.65.
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Let x,X. X30o be a random sample from Gamma(3, 2). Find an approximate probability that x is less than 6.3 (round off to third decimal place) 42300
0.933 is approximate probability that x is less than 6.3 (round off to third decimal place).
What is Probability?
Calculating or estimating how likely or "possible" something is to occur is the subject of probability.
Words like "certain," "impossible," or "probable" can be used to express the likelihood of an event occurring.
Probabilities in mathematics are always represented as fractions, decimals, or percentages with values ranging from 0 to 1.
Given X₁,X₂, ............X₃₀₀ are IID random samples from Xi ~ Gamma ( shape = 3 scale = 2 ) . The mean and variance are
E(Xi) = 3 * 2 = 6
Var (Xi) = 3 * 2² = 12
According to CLT,
X = X₁ + X₂ + ............+ X₃₀₀/300
has approximately normal distribution X ~ N( 6,12/300) So the probability,
P( X < 6.3 ) = P( Z < 6.3 - 6/√12/300
P ( X < 6.3 ) = P( Z < 1.5 )
P (X < 6.3 ) = Ф (1.5 )
P (X < 6.3 ) = 0.933
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help
Samantha is going to invest in an account paying an interest rate of 6.4%
compounded monthly. How much would Samantha need to invest, to the nearest
dollar, for the value of the account to reach $2,130 in 12 years?
Based on the given parameters, the value of the money to invest initially is $994.8
How to determine the amount of investment?The given parameters about the compound interest are
Principal Amount, P = $2130
Interest Rate, R = 6.4%
Time, t = 12
Number of times, n = 12 i.e. monthly
To calculate the amount, we have:
A = P + CI
Where
C = P(1 + R)^t
So, we have
A = P(1 + R)^t
This can be rewritten as
A = P(1 + R/n)^nt
Substitute the known values in the above equation
2130 = A * (1 + 6.4%/12)^(12*12)
So, we have
A = 2130/[(1 + 6.4%/12)^(12*12)]
Evaluate the expression
A = 994.8
Hence, the amount of the investment is $994.8
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A bacteria culture contains 2000 bacteria and doubles every half hour, find the size of the bacterial population after 5 hours
Find the values of x and y.
Answer:
x = y = 60
Step-by-step explanation:
You want the values of angles marked x° and y° in the given figure involving equilateral triangles and parallel lines.
TrianglesAll of the triangles in the figure are congruent equilateral triangles. Each of their internal angle measures is 60°.
x° = y° = 60°
__
Additional comment
The (inside) base angles of the outside two triangles are alternate interior angles with respect to the top two angles of the center triangle. Hence they are congruent. The center triangle is marked as equilateral, so its angles are all 60°. That means all of the acute angles in the figure are 60°.
Find the sum.
(4x² + x − 4) + (2x² + 4x + 5) =
20 POINTS PEASE
Answer:
6x² + 5x + 1
Step-by-step explanation:
(4x² + x − 4) + (2x² + 4x + 5)
1. Expand the brackets to get
4x² + x − 4 + 2x² + 4x + 5
2. Group common terms (terms with same power as well as the constant)
4x² + 2x² + x + 4x - 4 + 5
3. Simplify
4x² + 2x² = 6x²
x + 4x = 5x
-4 + 5 = 1
4. Solution is
6x² + 5x + 1