Therefore, the number of animals at the start of year 3 is approximately 408.04.
What is equation?An equation is a mathematical statement that shows the equality of two expressions, usually with an equal sign "=" in between them. An equation can contain variables, constants, and operators. The variables are represented by letters and can take on different values, while constants are fixed values that do not change. Operators include mathematical symbols like plus, minus, multiplication, and division, as well as exponents and logarithms.
Here,
We are given that the population at the start of year 1 (P1) is 400. Using the recurrence relation Pt + 1 = 1.01Pt, we can find the population at the start of year 2 (P2) and year 3 (P3) as follows:
P2 = 1.01 * P1 = 1.01 * 400 = 404
P3 = 1.01 * P2 = 1.01 * 404 = 408.04
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The equations were multiplied by constants and added together to determine that the value of x in the system of
equations is -4.
What is the value of y?
0-6
O-3
O 3
06
As a result the system of equation , y has a value that is roughly 4.33.
SYSTEM OF EQUATIONS: WHAT IS IT?A group of two or more equations with the same variables is referred to as a system of equations. The collection of numbers that simultaneously make all the equations true is the system's solution.
For instance, take into account the equations below:
2x + y = 5
x - y = 1
There are two equations with two variables in this system (x and y). The collection of numbers that simultaneously satisfy both equations is the system's solution. The answer in this situation is x = 2 and y = 1.
We may solve for y by substituting the value of x in the system of equations, which is -4, into one of the equations.
Consider the following two equations:
axe + by = c
dx + ey = f
We can solve for y by substituting the value of x into one of the equations if we know it.
As an illustration, if x = -4, then we have:
2x + 3y = 5
4x - 3y = -16
We can change one of the equations to read x = -4:
2(-4) + 3y = 5
When we simplify this equation, we obtain:
-8 + 3y = 5
8 is added to both sides to get at:
3y = 13
When we multiply both sides by 3, we get:
y = 13/3
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One hour later the right triangle is 15 inches long and 13 inches high and the base and height are changing at the same rate is there hypotenuse increasing or decreasing now?
The hypotenuse of the new triangle is increasing.
What theorem can be used to determine the triangle?The Pythagorean Theorem can be used to determine the new right triangle's hypotenuse length:
[tex]c^2 = a^2 + b^2\\c^2 = 13^2 + 15^2\\c^2 = 169 + 225\\c^2 = 394\\c ≈ 19.85[/tex]
Hence, the hypotenuse of the new right triangle measures roughly 19.85 inches in length.
Let h be the height of the triangle, and let b be the base of the triangle, both as functions of time t. We know that the base and height are changing at the same rate, so we can express this as:
dh/dt = db/dt = k, where k is a constant rate of change.
We can use the Pythagorean Theorem to express the length of the hypotenuse c as a function of h and b:
[tex]c^2 = h^2 + b^2[/tex]
Taking the derivative of both sides with respect to time t using the chain rule, we get:
2c(dc/dt) = 2h(dh/dt) + 2b(db/dt)
Simplifying and substituting the values for h, b, and c at time t = 0 (when the original triangle was given):
2c(dc/dt) = 2(7)(k) + 2(x)(k)
2c(dc/dt) = 14k + 10k
2c(dc/dt) = 24k
adding 2c to both sides' sums:
(12k/c) = (dc/dt)
For the new triangle, substitute the value of c as follows:
(dc/dt) = 12k/19.85
The sign of (dc/dt) will be decided by the sign of 1/c because k is positive and constant. Since we know that 1/c is positive and that c is positive, we can infer that (dc/dt) will also be positive because it will have the same sign as k. As a result, the new triangle's hypotenuse is growing.
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John is a 45 year old educator . He is living with his spouse and two children below 20 years . He earns a gross monthly salary R 21 400 . He contributes 7,5 % of his salary to the pension fund . He is the main member of a medical aid and his wife and children are his dependents . Get a copy of the medical tax credit rates from 2013 - 2023 and rebates from 2015 to 2023
Answer:
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Step-by-step explanation:
Medical tax credit rates and rebates are determined by the South African Revenue Service (SARS) and are subject to change from year to year. Therefore, I cannot provide you with a copy of the medical tax credit rates and rebates for the entire period of 2013 to 2023.
However, I can provide you with the following information:
The medical tax credit rate for the 2021/2022 tax year is R319 per month for the main member and the first dependent, and R215 per month for each additional dependent.
The medical tax credit rate for the 2020/2021 tax year was R310 per month for the main member and the first dependent, and R209 per month for each additional dependent.
The medical tax credit rate for the 2019/2020 tax year was R310 per month for the main member and the first dependent, and R209 per month for each additional dependent.
The medical tax credit rate for the 2018/2019 tax year was R303 per month for the main member and the first dependent, and R204 per month for each additional dependent.
The medical tax credit rate for the 2017/2018 tax year was R286 per month for the main member and the first dependent, and R192 per month for each additional dependent.
The medical tax credit rate for the 2016/2017 tax year was R286 per month for the main member and the first dependent, and R192 per month for each additional dependent.
The medical tax credit rate for the 2015/2016 tax year was R270 per month for the main member and the first dependent, and R181 per month for each additional dependent.
The medical tax credit rate for the 2014/2015 tax year was R257 per month for the main member and the first dependent, and R172 per month for each additional dependent.
As for rebates, I should note that the South African tax system works with a system of tax brackets, where the amount of tax you pay increases as your income increases. Tax rebates are a form of tax relief that is subtracted from the amount of tax you owe. The rebate amount changes annually, and for the 2021/2022 tax year, the rebate for individuals under 65 years of age is R15,714.
Again, it is important to note that these rates and rebates are subject to change from year to year, and it is recommended that you consult with a qualified tax professional for the most up-to-date information.
Find the common ratio of 1,3,9,-27 and the next three terms
Answer:
Step-by-step explanation:
The common ratio of this sequence is 3. Next 3 terms are -81, -243, and -729.
pls help due tmrw i want a good grade
find the slope of the line that that passes through each pair of points (8,-2) (4,-3)
Answer:
slope = [tex]\frac{1}{4}[/tex]
Step-by-step explanation:
calculate the slope m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (8, - 2 ) and (x₂, y₂ ) = (4, - 3 )
m = [tex]\frac{-3-(-2)}{4-8}[/tex] = [tex]\frac{-3+2}{-4}[/tex] = [tex]\frac{-1}{-4}[/tex] = [tex]\frac{1}{4}[/tex]
Write a system of equations to solve the following scenario. Twins Jon and Ron together earned $96,000 last year. Ron earned $8,000 more than three times what Jon earned. How much did each of the twins earn? Use the variable j to represent the amount Jon earned. Use the variable r to represent the amount Ron earned.
The amount Jon and Ron earn are $22,000 and $ 74,000 respectively.
What is word problem?Word problem is a mathematical problem expressed entirely in words typically used as an educational tool.This word problem is interpreted into mathematical equation or expression.
Represent the amount Jon earn by j and Ron earn by r
j+r = 96000 equation 1
r = 8000+ 3j equation 2
from equation 1
r = 96000-j
substitute 96000-j for r in equation 2
96000-j = 8000 +3j
collect like terms
96000-8000 = 3j +j
88,000 = 4j
j = 88000/4
j = $22,000
from equation 1
22000+r = 96000
r = 96000-22000
r = 74,000
Therefore the amount Jon and Ron earn are $22,000 and $ 74,000 respectively.
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Determine each ratio as a decimal to four places, then find the angle to the nearest whole number
The ratio as decimal is sin C = AB/BC = 8/17 = 0.4706. cos C = AC/BC = 15/17 = 0.8824 tan B = AB/AC = 8/15 = 0.5333. tan C = AB/AC = 8/15 = 0.5333. cos B = BC/AC = 17/15 = 1.1333. sin B = AC/BC = 15/17 = 0.8824.
What is unit circle?The origin (0, 0) of a coordinate plane serves as the center of the unit circle, which has a radius of 1. It is used in trigonometry to provide the values of trigonometric functions for angles of any measure, including sine, cosine, and tangent. The x-coordinate and y-coordinate of each point on the unit circle correspond to the cosine and sine values, respectively, for that point's specific angle measure. The unit circle is a popular visual tool for teaching students about the characteristics and behaviour of trigonometric functions.
Using the right triangle we have:
sin C = opposite/hypotenuse = AB/BC = 8/17 = 0.4706
cos C = adjacent/hypotenuse = AC/BC = 15/17 = 0.8824
tan B = opposite/adjacent = AB/AC = 8/15 = 0.5333
tan C = opposite/adjacent = AB/AC = 8/15 = 0.5333
cos B = adjacent/hypotenuse = BC/AC = 17/15 = 1.1333
sin B = opposite/hypotenuse = AC/BC = 15/17 = 0.8824
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mr.jones decided to invest in the stock market, but the market declined by an average of 2.5% each month during the first 6 months of his investment. If mr.jones initially invested $1,500, how much does he have remaining?
Step-by-step explanation:
If the stock market declined by an average of 2.5% each month during the first 6 months of Mr. Jones' investment, we can calculate the total percentage decrease using the following formula:
(1 - 0.025)^6 = 0.8938
This means that after 6 months, Mr. Jones' investment would have decreased by approximately 10.62% (1 - 0.8938 = 0.1062).
To calculate how much money Mr. Jones has remaining, we need to multiply his initial investment of $1,500 by the percentage decrease:
$1,500 x 0.1062 = $159.30
Therefore, Mr. Jones would have $1,500 - $159.30 = $1,340.70 remaining after the first 6 months of his investment in the stock market with an average decline of 2.5% each month.
three consecutive integers whose sum is -6. what are the numbers
Answer:
The numbers are -3, -2, and -1.
Step-by-step explanation:
Since -2 and -1 add up to -3, and -3+-3 = -6, and since the numbers are next to each other on your number line, well, then you got your answer. Here's how:
-3+-2+-1=-6
Which linear function represents a slope of 1/4?
Answer:
The second answer choice represents a slope of 1/4.
Step-by-step explanation:
Using the coordinates (8,5) and (0,3) you can do point slope form (y2-y1/x2-x1). In this problem that would translate to 5-3/8-0 which is 2/8 and can be simplified to 1/4.
1. They each walk 500 units to school. Who walks 500 feet, and who walks 500 yards? Explain your reasoning
Answer:
Below
Step-by-step explanation:
If they each walk in straight lines to the school Lin is closer than Elena
so Lin walks 500 ft and Elena walks 500 yards ( which is 1500 ft)
Please help, tomorrow is Math examination!
We know that the circumference (C) of a circle can be calculated using the formula:
C = 2πr, where r is the radius of the circle.
We are given that the circumference of the circle is 2.20 cm, so we can use this to solve for the radius (r):
C = 2πr
2.20 = 2πr
r = 2.20 / (2π)
r ≈ 0.350 cm
Therefore, the radius of the circle is approximately 0.350 cm.
To find the area (A) of the circle, we can use the formula:
A = πr^2
Substituting the value we found for r:
A = π(0.350)^2
A ≈ 0.385 cm^2
Therefore, the area of the circle is approximately 0.385 cm^2.
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Answer:
a) Radius of circle is 35 cm.b) Area of the Circle is 3850 cm²Step-by-step explanation:
Question :-
The circumference of the circle is 220 cm.
Find :
a) Radius
b) Area of the Circle
Solution :
a)
Circumference of circle = 2πr
[tex] \longrightarrow \: \: 220 = 2 \pi r \\ \\ \longrightarrow \: \: \frac{220}{2} = \pi r \\ \\ \longrightarrow \: \: 110 = \frac{22}{7} × r \\ \\ \longrightarrow \: \: r = 110 \times \frac{7}{22} \\ \\\longrightarrow \: \: r = \frac{770}{22} \\ \\ \longrightarrow \: \: r = 35 \: cm \\ [/tex]
b)
Area of circle = πr²
[tex]\longrightarrow \: \: \frac{22}{7} \times 35 \times 35 \\ \\ \longrightarrow \: \:22 \times 5 \times 35 \\ \\ \longrightarrow \: \:3850 \: {cm}^{2} \\ [/tex]
Hence,
a) Radius of circle is 35 cm.
b) Area of the Circle is 3850 cm²
An automobile company is running a new television commercial in five cities with approximately the same population. The following table shows the number of times the commercial is run on TV in each city and the number of car sales (in hundreds). Find the Pearson correlation coefficient r for the data given in the table. Round any intermediate calculations to no less than six decimal places, and round your final answer to three decimal places. r=
Pearson correlation coefficient r for the data is approximately 0.915. Since the value is positive and close to 1, we can say that there is a strong positive correlation between the number of times the commercial is run on TV and the number of car sales.
What is expression ?
In mathematics, an expression is a combination of numbers, symbols, and operators (such as +, -, ×, ÷) that represents a value or a quantity. It can be a single number or variable, or a combination of them, and can also include functions, parentheses, and other mathematical symbols. For example, 3x + 7 is an expression, where 3 and 7 are constants, x is a variable, and + is an operator. Expressions can be simplified, evaluated, or used as part of a larger mathematical statement or equation.
We can use the formula for Pearson correlation coefficient r to calculate it for the given data:
r = (nΣXY - ΣXΣY) / sqrt[(nΣX² - (ΣX)²)(nΣY² - (ΣY)²)]
where n is the number of observations, Σ represents the sum of the indicated values, and X and Y represent the two variables being correlated. Using the data from the table, we can calculate the following:
n = 5
ΣX = 25, ΣY = 25.8
ΣX² = 275, ΣY² = 288.68
ΣXY = 139.2
Substituting these values into the formula, we get:
[tex]$r = (5 * 139.2 - 25 * 25.8) \sqrt{[(5 * 275 - 25^2)(5 * 288.68 - 25.8^2)]} $[/tex]
r ≈ 0.915
Therefore, the Pearson correlation coefficient r for the data is approximately 0.915. Since the value is positive and close to 1, we can say that there is a strong positive correlation between the number of times the commercial is run on TV and the number of car sales.
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What is the equation for this graph?
Answer:
[tex]y=\frac{5}{2} x[/tex]
Step-by-step explanation:
Notice is a line, The equation for a line can be found as the following:
[tex]y-yo=m(x-xo)[/tex]
We need to find the slope.
First take two points of the graph
(0,0) the origin and (4,10) for example. Notice i do assume the X value is 4 for Y=10, since the cropping for the picture doesn't show the exact values for X .
Then the slope is given by equation:
[tex]m=(y-yo)/(x-xo)\\[/tex]
[tex]m=\frac{10-0}{4-0} =5/2[/tex]
then the equation is
[tex]y-0=\frac{5}{2} m(x-0)[/tex]
[tex]y=\frac{5}{2} x[/tex]
Answer:
y=5/2x or y=2.5x
Step-by-step explanation:
What is JL if KM=6?
Not really much context but that’s the whole question
when running a line, in a right-triangle, from the 90° angle perpendicular to its opposite side, we will end up with three similar triangles, one Small, one Medium and a containing Large one. Check the picture below.
[tex]\cfrac{x}{6}=\cfrac{6}{8}\implies \cfrac{x}{6}=\cfrac{3}{4}\implies x=\cfrac{18}{4}\implies x=\cfrac{9}{2} \\\\[-0.35em] ~\dotfill\\\\ 8+x\implies 8+\cfrac{9}{2}\implies \cfrac{25}{2}\implies 12\frac{1}{2}=JL[/tex]
Step-by-step explanation:
geometric mean theorem :
the height of a right-angled triangle to the Hypotenuse is the square root of the product of the parts of the Hypotenuse (as the height splits the Hypotenuse into 2 parts : JM and ML).
JL = JM + ML
in short
KM = sqrt(JM × ML)
6 = sqrt(8 × ML)
36 = 8 × ML
ML = 36/8 = 9/2 = 4.5
so,
JL = JM + ML = 8 + 4.5 = 12.5
One wall of a room measures 14 feet long and 8 feet
high. It contains a window 5 feet wide and 3.5 feet
high. The wall has an effective total R-value of 15.5.
Find the rate of heat flow through the wall when
the inside air temperature is 68 °F and the outside
temperature is 5 °F.
Answer: the rate of heat flow through the wall when the inside air temperature is 68 °F and the outside temperature is 5 °F is 257.42 BTU per hour.
Step-by-step explanation: The rate of heat flow through the wall can be found using the formula:
Rate of heat flow = (Temperature difference) / (Effective R-value)
The temperature difference is the difference between the inside and outside temperatures, which is:
Temperature difference = (68°F) - (5°F) = 63°F
The effective R-value of the wall is given as 15.5.
To calculate the total area of the wall, we first need to find the area of the window, which is:
Area of window = (width) x (height) = (5 ft) x (3.5 ft) = 17.5 square feet
The area of the wall without the window is:
Area of wall = (length) x (height) - Area of window
Area of wall = (14 ft) x (8 ft) - 17.5 square feet
Area of wall = 105.5 square feet
So, the rate of heat flow through the wall is:
Rate of heat flow = (Temperature difference) / (Effective R-value) x (Total area of the wall)
Rate of heat flow = (63°F) / (15.5) x (105.5 square feet)
Rate of heat flow = 257.42 BTU per hour
Write an equation of the line that passes through the point (–4, 6) with slope –4.
A. y−6=−4(x+4)
B. y+4=−4(x−6)
C. y−6=4(x+4)
D. y+4=4(x−6)
Answer:
Answer is A
Step-by-step explanation:
General equation of a line is given by;
[tex]{ \boxed{ \rm{y = mx + c}}}[/tex]
m is gradientc is y-intercept.From the question, m = -4
[tex]{ \rm{y = - 4x + c}}[/tex]
For (-4, 6)
[tex]{ \rm{6 = ( - 4 \times - 4) + c}} \\ { \rm{6 = 16 + c}} \\ { \rm{c = - 10}}[/tex]
Therefore;
[tex]{ \rm{y = - 4x - 10}} \\ { \colorbox{silver}{subtract \: 6 \: from \: both \: sides}} \\ { \rm{y + ( - 6) = ( - 4x - 10) + ( - 6)}} \\ \\ { \rm{y - 6 = - 4x - 16}} \\ \\ { \rm{y - 6 = - 4(x + 4)}}[/tex]
summation from n equals 2 to 6 of quantity 4 times n plus 5 end quantity period a 85 b 105 c 114 d 147
The value of the sum from n = 2 to n = 6 of 4n + 5 is given as follows:
b. 105.
How to obtain the numeric value of a function or of an expression?To obtain the numeric value of a function or of an expression, we substitute each instance of any variable or unknown on the function by the value at which we want to find the numeric value of the function or of the expression.
The expression of this problem is given as follows:
4n + 5.
The sum is the sum of the numeric values from n = 2 to n = 6, hence:
n = 2: 4(2) + 5 = 13.n = 3 -> 17.n = 4 -> 21.n = 5 -> 25.n = 6 -> 29.Hence the sum has the result given as follows:
13 + 17 + 21 + 25 + 29 = 105.
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PLSS i NEED help :/ Can u show me the method too? I'm really stuck
IS IT RIGHT??? OR I MISSED SOMETHING
It takes 32 pounds of seed to completely plant a 4 -acre field. How many acres can be planted per pound of seed?
Answer:
Step-by-step explanation:
To solve this problem, we need to find the number of acres that can be planted per pound of seed.
We know that it takes 32 pounds of seed to plant a 4-acre field. Therefore, the amount of seed needed to plant 1 acre is:
32 pounds / 4 acres = 8 pounds/acre
This means that for every pound of seed, we can plant:
1 acre / 8 pounds = 0.125 acres/pound
Therefore, we can plant 0.125 acres per pound of seed.
please help fast!! Given m∥n, find the value of x.
(4x+3) (8x-3)
A 6000-seat theater has tickets for sale at $26 and $40. How many tickets should be sold at each price for a sellout performance to generate a total revenue of $195.200?
In ΔWXY, the measure of ∠Y=90°, WY = 5, XW = 13, and YX = 12. What ratio represents the cosine of ∠X?
Answer:
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Check the picture below.
The total cost of producing a type of boat is given by C(x)=22000−40x+0.02x2
, where x is the number of boats produced. How many boats should be produced to incur minimum cost?
Answer:
To find the number of boats that should be produced to incur minimum cost, we need to find the value of x that minimizes the cost function C(x).
We can do this by taking the derivative of the cost function with respect to x, setting it equal to zero, and solving for x.
C(x) = 22000 - 40x + 0.02x^2
C'(x) = -40 + 0.04x
Setting C'(x) = 0, we get:
-40 + 0.04x = 0
0.04x = 40
x = 1000
Therefore, the number of boats that should be produced to incur minimum cost is 1000.
Step-by-step explanation:
Assume that 58% of people are left-handed. If we select 5 people at random, find the probability of each outcome described below, rounded to four decimal places:
a. There are some lefties ( ≥ 1) among the 5 people.
b. There are exactly 3 lefties in the group.
c. There are at least 4 lefties in the group.
d. There are no more than 2 lefties in the group.
e. How many lefties do you expect?
f. With what standard deviation?
Answer:
a. To find the probability that there are some lefties among the 5 people, we need to find the probability of the complement event, which is that there are no lefties among the 5 people. The probability of an individual being right-handed is 1 - 0.58 = 0.42. Therefore, the probability of none of the 5 people being left-handed is:
P(no lefties) = 0.42^5 = 0.0070 (rounded to four decimal places)
The probability of there being some lefties (≥ 1) is the complement of this:
P(some lefties) = 1 - P(no lefties) = 1 - 0.0070 = 0.9930 (rounded to four decimal places)
Therefore, the probability of there being some lefties among the 5 people is 0.9930.
b. To find the probability of there being exactly 3 lefties in the group, we can use the binomial probability formula:
P(exactly 3 lefties) = (5 choose 3) * (0.58)^3 * (0.42)^2
where (5 choose 3) = 10 is the number of ways to choose 3 people out of 5. Plugging in the values, we get:
P(exactly 3 lefties) = 10 * 0.58^3 * 0.42^2 = 0.3383 (rounded to four decimal places)
Therefore, the probability of there being exactly 3 lefties among the 5 people is 0.3383.
c. To find the probability of there being at least 4 lefties in the group, we can use the binomial probability formula again:
P(at least 4 lefties) = P(4 lefties) + P(5 lefties)
P(4 lefties) = (5 choose 4) * (0.58)^4 * (0.42)^1 = 0.2684
P(5 lefties) = (5 choose 5) * (0.58)^5 * (0.42)^0 = 0.1037
Adding these probabilities, we get:
P(at least 4 lefties) = 0.2684 + 0.1037 = 0.3721 (rounded to four decimal places)
Therefore, the probability of there being at least 4 lefties among the 5 people is 0.3721.
d. To find the probability of there being no more than 2 lefties in the group, we can use the binomial probability formula again:
P(no more than 2 lefties) = P(0 lefties) + P(1 lefty) + P(2 lefties)
P(0 lefties) = (5 choose 0) * (0.58)^0 * (0.42)^5 = 0.0022
P(1 lefty) = (5 choose 1) * (0.58)^1 * (0.42)^4 = 0.0344
P(2 lefties) = (5 choose 2) * (0.58)^2 * (0.42)^3 = 0.1866
Adding these probabilities, we get:
P(no more than 2 lefties) = 0.0022 + 0.0344 + 0.1866 = 0.2232 (rounded to four decimal places)
Therefore, the probability of there being no more than 2 lefties among the 5 people is 0.2232.
Step-by-step explanation:
HELP A construction company takes 1 7/8 hours to remove 2 3/4 of dirt what is the hit rate metrics ton per hour
Answer: To calculate the hit rate metric in tons per hour, we need to first calculate how many tons of dirt the construction company can remove in one hour.
To do this, we'll convert the given time and amount of dirt into a rate of dirt removal in tons per hour:
The company takes 1 7/8 hours to remove 2 3/4 tons of dirt
We can convert 1 7/8 hours to an improper fraction: 15/8 hours
We can convert 2 3/4 tons to an improper fraction: 11/4 tons
We can calculate the rate of dirt removal as the amount of dirt divided by the time: (11/4 tons) / (15/8 hours) = (11/4) ÷ (15/8) = (11/4) x (8/15) = 22/15 tons per hour
Therefore, the construction company has a hit rate metric of 22/15 tons per hour, which simplifies to 1.47 tons per hour (rounded to two decimal places).
Step-by-step explanation:
Look at the set of ordered pairs.
{(4, 6), (−7,−15), (13, 15), (−21, 8), (?, ?)}
Which of the following could replace the missing ordered pair to make the set not a function?
(−2, 4)
(21, 9)
(−7, 15)
(6, 8)
(13,−15)
The ordered pair that could replace the missing ordered pair to make the set not a function is given as follows:
(−7, 15).
When does a set of ordered pairs represent a function?A set of ordered pairs represents a function if every input (first component of the ordered pair) is associated with exactly one output (second component of the ordered pair). In other words, if no two ordered pairs in the set have the same first component but different second components.
Hence the ordered pair (-7,15) would make the relation not a function, as the set already has the ordered pair (-7,-15), in which the input 7 is already mapped to an output of -15, hence it cannot be mapped to an output of 15.
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The Rodriguez family and the Barnes family each used their sprinklers last summer. The Rodriguez family's sprinkler was used for 15 hours. The Barnes family's sprinkler was used for 35 hours. There was a combined total output of 1050 L of water. What was the water output rate for each sprinkler if the sum of the two rates was 50 L per hour?
Rodriguez family's sprinkler:
Barnes family's sprinkler:
The water output rate for the Rodriguez family's sprinkler was 35 L per hour, and the water output rate for the Barnes family's sprinkler was 15 L per hour (50 - 35) by solving the expression formed.
What is expression ?In mathematics, an expression is a combination of numbers, variables, and mathematical operations, such as addition, subtraction, multiplication, and division, that are written in a certain order. Expressions can be evaluated or simplified by following the rules of arithmetic or algebra. Examples of expressions include "2x + 3", "(a + b) / c", and "5^2 - 3".
According to given information :
Let's use the variable "r" to represent the water output rate for the Rodriguez family's sprinkler.
Then, the water output rate for the Barnes family's sprinkler can be represented as "50 - r", since the sum of the two rates is 50 L per hour.
Using the formula:
rate x time = amount of output, we can set up two equations:
r x 15 = amount of water output for the Rodriguez family
(50 - r) x 35 = amount of water output for the Barnes family
We also know that the total amount of water output is 1050 L, so we can set up a third equation:
r x 15 + (50 - r) x 35 = 1050
Now we can solve for "r".
First, simplify the third equation
15r + 1750 - 35r = 1050-20r = -700r = 35
Therefore, the water output rate for the Rodriguez family's sprinkler was 35 L per hour, and the water output rate for the Barnes family's sprinkler was 15 L per hour (50 - 35).
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Find all critical points for the function
4x + 6
x² + x + 1
on (-∞, ∞) and then list them (separated by commas) in the box below.
List of critical points:
f(x) =
The critical points for the function are [tex]x=-\frac{3+\sqrt{7}}{2},\:x=-\frac{3-\sqrt{7}}{2}[/tex]
How to detemine the critical points for the functionFrom the question, we have the following parameters that can be used in our computation:
f(x) = (4x + 6)/(x² + x + 1)
The critical points are the points where the derivative of f(x) equals 0 or undefined when the function is defined
When f(x) is differentiated, we have
f'(x) = 4/(x² + x + 1) - [(2x + 1)(4x + 6)/[(x² + x + 1)²]
Set to 0 and evaluate
4/(x² + x + 1) - [(2x + 1)(4x + 6)/[(x² + x + 1)²] = 0
So, we have
[(2x + 1)(4x + 6)/[(x² + x + 1)²] = 4/(x² + x + 1)
This gives
[(2x + 1)(4x + 6)/[(x² + x + 1)] = 4
Cross multiply
(2x + 1)(4x + 6) = 4x² + 4x + 4
12x² + 12x + 4x + 6 = 4x² + 4x + 4
12x² + 12x + 6 = 4x² + 4
Evaluate
8x² + 12x + 2 = 0
So, we have
4x² + 6x + 1 = 0
Using a graphing tool, we have
[tex]x=-\frac{3+\sqrt{7}}{2},\:x=-\frac{3-\sqrt{7}}{2}[/tex]
Hence, the critical points are [tex]x=-\frac{3+\sqrt{7}}{2},\:x=-\frac{3-\sqrt{7}}{2}[/tex]
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