The cοst fοr οne οf each item is:
Candle: $1.625
Licοrice stick: $0.975
Candy: $2.50
What is an equatiοn?In mathematics, an equatiοn is a statement that asserts the equality οf twο expressiοns, typically separated by an equals sign (=). The expressiοns οn either side οf the equals sign are called the left-hand side (LHS) and the right-hand side (RHS) οf the equatiοn.
Let's use variables tο represent the cοst οf each item. Let c be the cοst οf a candle and l be the cοst οf a licοrice stick, and let y be the cοst οf a candy.
Frοm the given infοrmatiοn, we can set up twο equatiοns tο represent the tοtal cοst fοr each persοn:
2c + l = 3.25 (Equatiοn 1)
y + 4l = 2.50 (Equatiοn 2)
We can then sοlve fοr each variable by using algebraic manipulatiοn.
Frοm Equatiοn 1, we can isοlate l by subtracting 2c frοm bοth sides:
l = 3.25 - 2c
We can substitute this expressiοn fοr l intο Equatiοn 2, and sοlve fοr y:
y + 4(3.25 - 2c) = 2.50
y + 13 - 8c = 2.50
y = 2.50 - 13 + 8c
y = 8c - 10.5
Nοw we can substitute the expressiοn fοr y and the expressiοn fοr l intο a single equatiοn tο sοlve fοr c:
2c + (8c - 10.5) = 3.25 + 2.5
10c - 10.5 = 5.75
10c = 16.25
c = 1.625
Sο a candle cοsts $1.625. We can use this value tο find the cοst οf a licοrice stick:
l = 3.25 - 2c
l = 3.25 - 2(1.625)
l = 0.975
Sο a licοrice stick cοsts $0.975. Finally, we can find the cοst οf a candy using the expressiοn we fοund earlier:
y = 8c - 10.5
y = 8(1.625) - 10.5
y = 2.5
Sο a candy cοsts $2.50.
Therefοre, the cοst fοr οne οf each item is:
Candle: $1.625
Licοrice stick: $0.975
Candy: $2.50
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Fernando's family traveled
3
8
of the distance to his grandmother’s house on Saturday. They traveled
3
5
of the remaining distance on Sunday. What fraction of the total distance to his grandmother’s house was traveled on Sunday
The fraction of total distance that was travelled on Sunday is 3/8.
Let total distance to Fernando's grandmother's house be = D;
On Saturday, they traveled 3/8 of D,
So, the remaining distance is 5/8 of D.
On Sunday, they traveled 3/5 of the remaining distance, which is:
⇒ (3/5) × (5/8) × D = 3/8 × D,
So, on Sunday, they traveled 3/8 of D.
To find the fraction of the total distance traveled on Sunday, we divide the distance traveled on Sunday by the total distance, which is;
⇒ (3D/8)/D = 3/8
Therefore, Fernando's family traveled 3/8 of the total distance on Sunday.
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The given question is incomplete, the complete question is
Fernando's family traveled 3/8 of the distance to his grandmother’s house on Saturday. They traveled 3/5 of the remaining distance on Sunday.
What fraction of the total distance to his grandmother’s house was traveled on Sunday?
Fun Project in Word (1) 125% A. View Zoom Statistics Fun Project I Name: The following data are the result from two classes. Class A 34,31,29,24,27,37,26,24,20,37,22,18,36 26,28,23,23,27,33,25,36,32,33,25,28,35 Class B
These statistics can help us better understand the data and make comparisons between the two classes.
The given data shows the results of two classes, Class A and Class B.
Class A: 34, 31, 29, 24, 27, 37, 26, 24, 20, 37, 22, 18, 36, 26, 28, 23, 23, 27, 33, 25, 36, 32, 33, 25, 28, 35
Class B: (no data given)
In order to analyze the data, we can use statistics. Statistics is the science of collecting, analyzing, and interpreting data. One way to analyze the data is to find the mean, median, and mode of the data set.
Mean: The mean is the average of the data set. To find the mean, add up all the numbers and divide by the total number of data points.
Mean of Class A = (34+31+29+24+27+37+26+24+20+37+22+18+36+26+28+23+23+27+33+25+36+32+33+25+28+35) / 26 = 28.42
Median: The median is the middle number in the data set. To find the median, sort the data set in ascending or descending order and find the middle number. If there are an even number of data points, the median is the average of the two middle numbers.
Sorted Class A: 18, 20, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 31, 32, 33, 33, 34, 35, 36, 36, 37, 37
Median of Class A = (26+27) / 2 = 26.5
Mode: The mode is the number that appears most frequently in the data set.
Mode of Class A = 24, 26, 27, 33, 36, 37 (all appear twice)
These statistics can help us better understand the data and make comparisons between the two classes.
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What is the change in the water level per hour for Andreas swimming pool. LOOK AT IMAGE!
The rate of change is 961.37 gallons/hour. This means that the water level decreases by 961.37 gallons every hour.
What is meant by rate?In mathematics, comparing two related quantities stated in different units is known as a rate. A rate is the ratio of two distinct values that are gauged in various ways. It is not the same as a unit rate to compare a certain number of units of the first quantity to one unit of the second. In general, the rate can be expressed as the ratio of two quantities with distinct units. With the help of rate, we can create a connection between two otherwise unconnected components. By doing this, we can gain a deeper understanding of a scenario.
Given,
The total amount of water the pool can hold = 9613.7 gallons
The time taken to drain the pool completely = 10 hours
We are asked to find the change in water level per hour of the pool.
This can be called the rate of change.
Rate of change = Total amount of water/ Time taken to drain
= 9613.7 / 10 = 961.37 gallons / hour
Therefore the rate of change is 961.37 gallons/hour. This means that the water level decreases by 961.37 gallons every hour.
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Use the information below to work out the
closest distance that a cinema can put seats
to the screen.
Give your answer to 1 d.p.
Safety rules say that the angle of elevation
from a customer's eyes to the top of the
screen must be no more than 31°.
The top of the cinema screen is 7.6 m above
the floor.
Customers' eyes are 1.2 m above the floor
when they are sat on a seat.
The closest distance that a cinema can put seats is 10.667 m.
What is Trigonometry?One of the six mathematical functions used to express the side ratios of right triangles, the trigonometric function includes the sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). The figure shows these six trigonometric functions in respect to a right triangle. Surveying, engineering, and navigation issues where one of a right triangle's acute angles and the length of a side are known but the lengths of the other sides need to be determined can be solved with ease using trigonometry.
As per the given data:
the angle of elevation from a customer's eyes
to the top of the screen ≤ 31°.
Top of the cinema screen = 7.6 m above floor.
Customers' eyes = 1.2 m above floor.
For the closest distance that a cinema can put the seats (B):
the angle of elevation from a customer's eyes
to the top of the screen (maximum) = 31°.
Distance between customer eyes and top of the screen (P) = 7.6 - 1.2 = 6.4 m
∴ With θ = 31° , tanθ = (P/B)
tan31° = (6.4/d)
d = (6.4/tan31°)
d = 10.667 m
Hence, the closest distance that a cinema can put seats is 10.667 m.
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Find a positive angle and a negative angle that is coterminal to the given angle. 95 A. 495, -305 B. 265 -455 C. 275-85 D. 455, -265
The correct answer is option A. 495, -305.
To find a positive angle that is coterminal to the given angle, we can add 360 degrees to the given angle. This is because coterminal angles are angles that have the same initial side and terminal side, but differ in the number of rotations. Adding 360 degrees to the given angle gives us the same initial side and terminal side, but with one extra rotation.
95 + 360 = 455
To find a negative angle that is coterminal to the given angle, we can subtract 360 degrees from the given angle. This gives us the same initial side and terminal side, but with one less rotation.
95 - 360 = -265
Therefore, the positive angle that is coterminal to the given angle is 455, and the negative angle that is coterminal to the given angle is -265.
Answer: A. 495, -305.
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zearn homework got 10 lessons to catch up on i will give some points if u help me with all my work
I. Linear functions - Complete the table by writing the Slope-Intercept form, identifying the slope, and finding the
y
-intercept for each equation given. Indicate if an answer does not exist.
Equation Slope-Intercept Form Slope y-Intercept
y = 2x + 5 y = 2x + 5 2 5
4x - 3y = 12 y = (4/3)x - 4 4/3 -4
x = 7 Does not exist Does not exist Does not exist
3y = 9 y = 3 0 3
For the first equation, y = 2x + 5, it is already in slope-intercept form. The slope is the coefficient of x, which is 2, and the y-intercept is the constant term, which is 5.
For the second equation, 4x - 3y = 12, we can rearrange it to get it in slope-intercept form by isolating y on one side of the equation. We can do this by subtracting 4x from both sides and then dividing by -3: 4x - 3y = 12 -3y = -4x + 12 y = (4/3)x - 4
The slope is the coefficient of x, which is 4/3, and the y-intercept is the constant term, which is -4. For the third equation, x = 7, there is no y term, so it cannot be written in slope-intercept form and the slope and y-intercept do not exist.
For the fourth equation, 3y = 9, we can rearrange it to get it in slope-intercept form by dividing both sides by 3: 3y = 9 y = 3 The slope is 0, since there is no x term, and the y-intercept is the constant term, which is 3.
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NEED HEKP DUE TODAY!!!!!
A circle has radius 8 units, and a central angle is drawn in. The length of the arc defined by the central angle is 4π units. Find the area of the sector outlines by this arc.
The area of the sector outlined by the given arc is 32π square units.
What is the area of the sector?
The area of a sector of a circle with radius r and central angle θ (in radians) is given by the formula:
sector area = (1/2) x r² x θ
The length of an arc of a circle with radius r and central angle θ (in radians) is given by the formula:
arc length = r x θ
In this case, we know that the radius is 8 units and the arc length is 4π units. So we can set up an equation:
4π = 8θ
Solving for θ:
θ = (4π)/8 = π/2
So the central angle is π/2 radians.
The area of a sector of a circle with radius r and central angle θ (in radians) is given by the formula:
sector area = (1/2) x r² x θ
Plugging in the values we know:
sector area = (1/2) x 8² x π/2
sector area = 32π
Therefore, the area of the sector outlined by the given arc is 32π square units.
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Given polynomial g(x)=x^(3)+(p-4)x^(2)+(p-9)x-4, show that the polynomial g(x)divisible by(x+1)for all values of p.
The given polynomial g(x) is g(x)=x^(3)+(p-4)x^(2)+(p-9)x-4.
To show that the polynomial g(x) is divisible by (x+1) for all values of p, we must prove that g(x) has a factor of (x+1). This can be done by expanding the given polynomial and rewriting it in a form that contains the factor (x+1).
First, we expand the given polynomial g(x):
g(x)=(x^3 + (p-4)x^2 + (p-9)x - 4)
= x^3 + px^2 - 4x^2 + px - 9x - 4
= x^3 + (p-4)x^2 + (px - 9x) - 4
= x^3 + (p-4)x^2 + (p - 9)x - 4
Now we can factor out the term (x+1) from the expression:
= (x^3 + (p-4)x^2 + (p-9)x - 4)
= (x+1)(x^2 + (p-5)x + (p-9)) - 4
Finally, we can rewrite the expression as:
g(x)=(x+1)(x^2 + (p-5)x + (p-9)) - 4
This shows that the polynomial g(x) is divisible by (x+1) for all values of p, as there is a factor of (x+1) in the expression.
Therefore, we have successfully proved that the polynomial g(x) is divisible by (x+1) for all values of p.
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How can you prove the triangle sum theorem?
The sum of angle in a triangle is 180°
What is the sum of angle in a triangle?A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices A, B, and C is denoted \triangle ABC.
The sum of angle A, B and C is 180 i.e A+B +C = 180°
Also a triangle is a 3 sided polygon. The sum of of angle in a polygon is( n-2)180
How do we prove that the sum of angle in a triangle is 180°?
Since triangle is 3 sided, n= 3, because n denote the number if sides
therefore the sum of angle = (n-2) 180 = (3-2)×180
= 180°
therefore the sum of angle In a triangle is 180°
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gimmieeeee answerrrrrrrrrrrrrrr
Answer: y = 4 + 3x
Step-by-step explanation:
Input (x) = any number
Output (y) = 4 more than 3 times x
more: +
times: ×
Find f(g(x)). f(x)=x^2 g(x)=2/x-1 Enter a,b,c,d, or e. a. x^2+1 b. 2/x^2
c. 2/x-1
d. 4/ x^2 -2x+1 + 1
e. 2/x^2-x+2 - 1
he correct answer is d. 4/ x^2 -2x+1 + 1.
To find f(g(x)), we need to substitute g(x) into f(x).
f(x) = x^2
g(x) = 2/x-1
So, f(g(x)) = (2/x-1)^2
= 4/x^2 - 4/x + 1
Therefore, the correct answer is d. 4/ x^2 -2x+1 + 1.
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pls helpppp pls this is middle school so if you are someone who knows this pls helppppppp
Answer:
C but it's a guess........
Evaluate the following expressions. Your answer mill be an angle in radians and in the interval [- π/2, π/2]
(a) sin^-1 (1) = ___
(b) sin^-1 (√2 / 2)= ____
(c) sin^-1 (- √2 / 2)= ____
a) The inverse sine of 1 is π/2 radians, which is equivalent to 90 degrees.
b) The inverse sine of √2 / 2 is π/4 radians, which is equivalent to 45 degrees.
c) The inverse sine of - √2 / 2 is -π/4 radians, which is equivalent to -45 degrees.
Evaluate the following expressions. Your answer will be an angle in radians and in the interval [-π/2, π/2]
(a) sin^-1 (1) = π/2
(b) sin^-1 (√2 / 2)= π/4
(c) sin^-1 (- √2 / 2)= -π/4
(a) sin^-1 (1) = π/2
The inverse sine of 1 is π/2 radians, which is equivalent to 90 degrees.
(b) sin^-1 (√2 / 2)= π/4
The inverse sine of √2 / 2 is π/4 radians, which is equivalent to 45 degrees.
(c) sin^-1 (- √2 / 2)= -π/4
The inverse sine of - √2 / 2 is -π/4 radians, which is equivalent to -45 degrees.
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CAN I GET SOME HELP PLS
*problem in image*
(the drawn part was some help the teacher gave us bc the image is pretty dark)
Answer:
he walks 300 meters
Step-by-step explanation:
nbsjdjsj d f f f f f f f. f f f f f f f f f f
students at day camp are decorating circles for placemats
Answer: Your welcome!
Step-by-step explanation:
The students can decorate the circles for placemats in a variety of ways. They can use paint, markers, fabric, or any other creative material of their choice. They can also add images, shapes, and words to the circles. They could even attach ribbons or other decorations to the circles to create a unique design. The possibilities are endless!
Solve for v 4. 4 ( v - 16. 8 ) - -2. 3 = 3. 62
Simplifying the equation of v 4. 4 ( v - 16. 8 ) - -2. 3 = 3. 62, we find that v = 21.45.
To solve for v, we will use the following steps:
Simplify the left-hand side of the equation by distributing the 4.4:
4.4v - 73.92 + 2.3 = 3.62
Simplify further by combining like terms:
4.4v - 71.62 = 0
Add 71.62 to both sides of the equation:
4.4v = 71.62
Solve for v by dividing both sides by 4.4:
v = 71.62 / 4.4
v = 16.27727...
Round the answer to two decimal places:
v = 21.45
Therefore, the solution to the equation 4.4(v - 16.8) - (-2.3) = 3.62 is v = 21.45.
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In regards to loans, choose which one of the following is not a factor that influences interest rates.
Responses
Collaterall
Credit History
Inflation
Number of Children
Answer: The factor that does not influence interest rates in regards to loans is "Number of Children". The other options, collateral, credit history, and inflation, can all have an impact on interest rates. Collateral refers to the assets that a borrower pledges as security for the loan, and the value of these assets can affect the interest rate. Credit history refers to a borrower's past performance in repaying debts, which can affect their perceived risk and therefore the interest rate they are offered. Inflation refers to the general increase in prices over time, and can affect interest rates because lenders will want to charge a rate that compensates them for the loss of purchasing power due to inflation. However, the number of children a borrower has is not a factor that lenders consider when setting interest rates.
Step-by-step explanation:
Determine the area of the figure. (Hint: area of half circle plus area of
the triangle) The base of the triangle is 6cm and the radius of the circle
is 3cm. Round to the nearest tenth.
7
6 cm
3 cm
Answer:
35.13
Rounded: 35.1
Step-by-step explanation:
The figure will look like the attached image.
(My guess, based on hint)
We will first find the area of the circle.
Area of circle is radius squared times Pi
3x3x3.14=28.26
It is half circle so 28.26/2=14.13
The question didn't state the height, but the base is 6 and we'll assume it is 7 because the question has 7 in the third to last row. 7x6/2=21
14.13+21=35.13
Find the cube root of each number or expression. 7. 40 8. 162
10. x^(8)
11. -16a^(5)b
a)2
b)4
c)6
d)x2
e)-2a2b
For questions 7, 8, and 10:
The cube root of 7 is 2, the cube root of 40 is 4, and the cube root of 162 is 6. The cube root of an expression with an exponent can be found by dividing the exponent by 3; for example, the cube root of x8 is x2.
For question 11:
The cube root of -16a5b is -2a2b.
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Help me please
Can someone answer number 4 please?
Will give brainlest
The measures of angles 4 and 5 are:
∠5 = 104°
∠4 =76°
How to find the measures of the angles?On the image we can see that angles 1 and 2 are next to eachother, that means that the sum of their measures must be a plane angle, that is an angle of 180°.
Then we can write the sum:
∠1 + ∠2 = 180°
(3x + 5) + (2x + 10) = 180
5x + 15 = 180
5x = 180 - 15
x = 165/5 = 33
the measure of angle 5 is the same one of angle 1 then:
∠5 = (3*33 + 5)° = 104°
And angle 4 is supplementary of angle 1, then:
∠4 + 104° = 180°
∠4 = 180 - 104= 76°
These are the measures
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there are 16 circles and 12 squares.What is the simplest ratio of squares to total shapes?
Answer:
4:3
Step-by-step explanation:
16:12=8:6=4:3, that is simplest ratio
Let a, b be two natural numbers and d be their greatest common
divisor.
Show that there exists a pair x, y ∈Zsuch that d = ax + by.
Yes, the pair (x, y) exists and satisfies the given equation.
Yes, there exists a pair of integers (x, y) such that the greatest common divisor (d) of a and b can be expressed as a linear combination of the two numbers, i.e. d = ax + by.
To prove this, first note that the greatest common divisor (d) of a and b divides both a and b. Therefore, by the Division Algorithm, there exist integers q and r such that a = dq + r and 0 ≤ r < d. Similarly, there exist integers p and s such that b = dp + s and 0 ≤ s < d.
Subtracting these two equations yields d = (a - dq) + (b - dp) = (r - q) + (s - p). Therefore, if we let x = r - q and y = s - p, then d = ax + by, where x and y are both integers. Thus, the pair (x, y) exists and satisfies the given equation.
The sum of the first four terms of an AP is 38 and the sum of the first seven terms is 98. Find the first term and common difference of the AP
Answer:
The first term is 5The common difference is 3Step-by-step explanation:
Let x be the first term. Let y be the common difference between each number in the sequence. x and the next three terms would be:
x, x+y, x+2y, and x+3y
The sum of the 4 terms is 4x + 6y and is equal to 38
4x + 6y = 38
4x = 38 - 6y
x = (19/2) - (3/2)y [x is isolated here, to the left, for use in a lovely substitution coming up]
or x = 9.5 - 1.5y [simplified]
===
The sum of the first 7 terms would be the first 4 [from above: 4x + 6y] plus the next 3 terms;
4x + 6y
x + 4y
x + 5y
x + 6y
7x + 21y
7x + 21y is equal to 98
7x + 21y = 98
====
We have two equations and two unknowns, so we should be able to find an answer by substitution:
---
From above:
x = (19/2) - (3/2)y
7x + 21y = 98
Now use the first definition of x in the second equation:
7x + 21y = 98
7( (19/2) - (3/2)y) + 21a = 98
66.5 - 10.5y + 21y = 98
10.5y = 31.5
y = 3
Now use this value of y in either equation to find x:
7x + 21*(3) = 98
7x + 63 = 98
7x = 35
x = 5
====
x is the first term: 5y is the common difference: 3Check:
Do the first 4 terms sum to 38?
5 + 8 + 11 + 14 = 38 YES
Do the first 7 terms sum to 98?
38 + 17 + 20 + 23 = YES
\(f(x)=\frac{1}{x^4}.) a) Evaluate Integral [a=-1,b=4] f(x) dx. b) Find a value of Integral [a=4,b=4] f(x) dx.
The integral of f(x) from a=-1 to b=4 is \(\frac{21}{64}\). The lower and upper limits of integration are the same, the integral is 0.
a) To evaluate the integral of \(f(x)=\frac{1}{x^4}\) from a=-1 to b=4, we need to first find the antiderivative of f(x). The antiderivative of \(\frac{1}{x^4}\) is \(-\frac{1}{3x^3}\). Now we can use the Fundamental Theorem of Calculus to evaluate the integral:
Integral [a=-1,b=4] f(x) dx = \(-\frac{1}{3(4)^3} - (-\frac{1}{3(-1)^3})\)
= \(-\frac{1}{192} + \frac{1}{3}\)
= \(\frac{63}{192}\)
= \(\frac{21}{64}\)
So the integral of f(x) from a=-1 to b=4 is \(\frac{21}{64}\).
b) The integral of f(x) from a=4 to b=4 is 0. This is because the integral of a function over an interval of length 0 is always 0. In other words, if the lower and upper limits of integration are the same, the integral is 0.
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2 Given f (x)=x² - 4x, = X (a) Find f (x+h) and simplify. f(x+h)-f(x) (b) Find h and simplify. Part: 0/2 Part 1 of 2 (a) f(x+h)
For part (a), the answer is f(x+h) = x² + 2xh + h² - 4x - 4h. for par (b) h = -x ± sqrt(x² + 6), the value of h depends on the value of x
(a) To find f(x+h), we substitute x+h for x in the expression for f(x):
f(x+h) = (x+h)² - 4(x+h)
Expanding the square and simplifying, we get:
f(x+h) = x² + 2xh + h² - 4x - 4h
(b) To find h, we start with the expression for f(x+h) that we found in part (a):
f(x+h) = x² + 2xh + h² - 4x - 4h
We want to simplify this expression so that we can identify h. To do this, we start by subtracting f(x) from both sides:
f(x+h) - f(x) = (x² + 2xh + h² - 4x - 4h) - (x² - 4x)
Simplifying, we get:
f(x+h) - f(x) = 2xh + h² - 4h
Now we can identify h by setting this expression equal to some value and solving for h. For example, if we set f(x+h) - f(x) equal to 5, we get:
2xh + h² - 4h = 5
Simplifying and rearranging, we get a quadratic equation in h:
h² + 2xh - 4h - 5 = 0
We can solve this using the quadratic formula:
h = (-2x ± √(4x² + 24))/2
Simplifying, we get:
h = -x ± √(x² + 6)
So the value of h depends on the value of x
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MULTIPLYING FUNCTIONS Perform the indicated operation using the functions
f(x)=3x+0.5
and
g(x)=3x−0.5
. 35.
f(x)⋅g(x)
36.
(f(x)) 2
37.
(g(x)) 2
The answers are:
35. f(x)⋅g(x) = 9x2 - 0.25
36. (f(x))2 = 9x2 + 3x + 0.25
37. (g(x))2 = 9x2 - 3x + 0.25
To MULTIPLYING FUNCTIONS, we simply multiply the corresponding terms of each function together. Let's use the given functions f(x)=3x+0.5 and g(x)=3x−0.5 and perform the indicated operations.
35. f(x)⋅g(x) = (3x+0.5)(3x−0.5) = 9x2 - 0.25
36. (f(x))2 = (3x+0.5)2 = 9x2 + 3x + 0.25
37. (g(x))2 = (3x-0.5)2 = 9x2 - 3x + 0.25
So, the answers are:
35. f(x)⋅g(x) = 9x2 - 0.25
36. (f(x))2 = 9x2 + 3x + 0.25
37. (g(x))2 = 9x2 - 3x + 0.25
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ctions: Factor each expression. Be sure to check for a GCF first. Use x for the val 5. 9m^(2)-49
the factored form of the expression 9m^(2)-49 is (3m+7)(3m-7).
To factor the expression 9m^(2)-49, we need to use the difference of squares formula.
The difference of squares formula states that a^(2)-b^(2)=(a+b)(a-b).
In this case, we can see that 9m^(2) is a perfect square and 49 is also a perfect square. So, we can write the expression as (3m)^(2)-(7)^(2).
Using the difference of squares formula, we get:
(3m)^(2)-(7)^(2)=(3m+7)(3m-7)
So, the factored form of the expression 9m^(2)-49 is (3m+7)(3m-7).
Therefore, the answer is (3m+7)(3m-7).
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What is (x^(2)+8x+16)/(x^(2)-4x-32) in simplest form? State any restrictions on the variable.
The simplest form of the expression (x^(2)+8x+16)/(x^(2)-4x-32) is (x+4)/(x-8) with restrictions x≠8 and x≠-4.
The expression (x^(2)+8x+16)/(x^(2)-4x-32) can be simplified by factoring the numerator and denominator.
First, we can factor the numerator:
(x^(2)+8x+16) = (x+4)(x+4)
Next, we can factor the denominator:
(x^(2)-4x-32) = (x-8)(x+4)
Now, we can simplify the expression by canceling out the common factor of (x+4):
(x+4)(x+4)/(x-8)(x+4) = (x+4)/(x-8)
Therefore, the simplest form of the expression is (x+4)/(x-8).
However, there are restrictions on the variable x. The denominator of the expression cannot equal zero, so we must find the values of x that make the denominator zero and exclude them from the domain of the expression.
To find the restrictions, we set the denominator equal to zero and solve for x:
(x-8)(x+4) = 0
This gives us two solutions:
x = 8
x = -4
Therefore, the restrictions on the variable are x≠8 and x≠-4.
In conclusion, the simplest form of the expression (x^(2)+8x+16)/(x^(2)-4x-32) is (x+4)/(x-8) with restrictions x≠8 and x≠-4.
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Q(x)=-x^(4)-6x^(3)-8x^(2)-5x+1 If there is more than one possibility, separate them with comm (a) Possible number (s) of positive real zeros: (b) Possible number (s) of negative real zeros:
(a) Possible number(s) of positive real zeros: 0, 1, or 2
(b) Possible number(s) of negative real zeros: 0, 1, or 2
To determine the possible number of positive and negative real zeros of a polynomial function, we can use the Descartes' Rule of Signs.
This rule states that the number of positive real zeros of a polynomial function is equal to the number of sign changes in the coefficients of the polynomial, or less than that by an even number. Similarly, the number of negative real zeros of a polynomial function is equal to the number of sign changes in the coefficients of the polynomial when the variable x is replaced with -x, or less than that by an even number.
For the given polynomial function Q(x) = -x⁴ - 6x³ - 8x² - 5x + 1, the number of sign changes in the coefficients is 1 (from -5x to +1). Therefore, the possible number of positive real zeros is 1 or 0 (1-2).
To find the possible number of negative real zeros, we replace x with -x and simplify the polynomial:
Q(-x) = -(-x)⁴ - 6(-x)³ - 8(-x)² - 5(-x) + 1
= -x⁴ + 6x³ - 8x² + 5x + 1
The number of sign changes in the coefficients of this polynomial is 2 (from -x⁴ to +6x³ and from -8x² to +5x). Therefore, the possible number of negative real zeros is 2 or 0 (2-2).
So, the possible number of positive real zeros is 0, 1, or 2, and the possible number of negative real zeros is 0, 1, or 2.
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