1. The x-intercepts are x=3 and x=1. Since 3 is greater than 1, the greatest x-intercept is x=3.
2. The y-intercept is -8.
3. g(x) has the greatest y-intercept.
What are the responses to other questions?1. The x-intercept of a function is found by setting f(x) = 0 and solving for x. So for f(x) = 2x²-8x+6, we have:
0 = 2x²-8x+6
0 = x²-4x+3
0 = (x-3)(x-1)
Therefore, the x-intercepts are x=3 and x=1. Since 3 is greater than 1, the greatest x-intercept is x=3.
2. The y-intercept of a function is found by setting x = 0 and evaluating f(x). So for f(x) = -x²-6x-8, we have:
f(0) = -(0)²-6(0)-8 = -8
Therefore, the y-intercept is -8.
For g(x), we have:
g(0) = 2(0)²+16(0)+31 = 31
Therefore, g(x) has the greatest y-intercept.
3. Similar to question 2, we find the y-intercept by setting x=0 and evaluating f(x):
f(0) = -(0)²-8(0)-12 = -12
Therefore, the y-intercept is -12.
4. To find the x-intercepts of f(x) = (x+8)²-4, we set f(x) = 0:
0 = (x+8)²-4
4 = (x+8)²
±2 = x+8
x = -8 ± 2
Therefore, the x-intercepts are x=-6 and x=-10. Since -6 is greater than -10, the greatest x-intercept is x=-6.
5. The axis of symmetry of a quadratic function in the form f(x) = ax²+bx+c is given by x = -b/(2a).
For f(x) = -2x²-12x-16, we have a = -2 and b = -12, so the axis of symmetry is x = -(-12)/(2*-2) = 3.
For g(x) = 2x²+16x+31, we have a = 2 and b = 16, so the axis of symmetry is x = -16/(2*2) = -4.
Therefore, f(x) has the greatest axis of symmetry.
6. To find the y-intercept, we set x=0 and evaluate the functions:
f(0) = 2(0-3)²+1 = 19
g(0) = 2(0+3)²+2 = 20
Therefore, g(x) has the greatest y-intercept.
7. To find the x-intercept, we set f(x) = 0:
0 = -(x+1)²+1
1 = (x+1)²
±1 = x+1
x = -1 ± 1
Therefore, the x-intercepts are x=-2 and x=0. Since -2 is greater than 0, g(x) has the greatest x-intercept.
8. Similar to question 6, we find the y-intercept by setting x=0 and evaluating the functions:
f(0) = -2(0+2)²+2 = -6
g(0) = 2(0+3)²-4 = 14
Therefore, g(x) has the greatest y-intercept.
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An insect population after x months can be modeled by the function g(x)=18(1.3)^x. Which statement is the best interpretation of one of the values in this function?
The base, which is equal to 1.3, is one of the values in the equation g(x)=18(1.3)x.
How are values of a function determined?The monthly growth rate of the insect population is represented by this number. Since 1.3 is 1 + 30% represented as a decimal, the population is specifically growing by 30% each month. When the growth rate is compounded each month, this indicates that the insect population is expanding quickly over time.
If we enter x=3 into the function, for instance, we obtain g(3)=18(1.3)3=18(2.197)=39.546. This indicates that the anticipated bug population after three months is 39,546. Pest control is frequently required to preserve ecosystem balance since this exponential growth might, if unchecked, result in a very huge insect population.
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EASY 15 POINTS PLEASE HELP ME!!!!!!
Answer: B
Step-by-step explanation:
(7x^3-10x^2- 5)-(2x^3+7x^2-9)
= 5x^3 -3x^2-14
4. The middle school is located at (0, 0) on
a coordinate plane. Town Hall is located
4 miles directly east of the middle
school. The fire station is located 2 miles
directly north of Town Hall.
How many miles long is a straight
line between the school and the fire
station? Round to the nearest tenth.
The distance between the middle school and the fire station is 4.5 miles
What is Pythagoras theorem?Pythagoras theorem states that, in a right triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides.
The distance between the middle school and the fire station is is the hypotenuse
therefore;
c² = 2² + 4²
c² = 4 + 16
c² = 20
c = √20
c = 4.5 miles(nearest tenth)
therefore the distance between the middle school and fire station is 4.5 miles
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Complete the identity (sin x +cos x) ^2
Please please help
Step-by-step explanation:
Using perfect square trinomial rules,
[tex](a + b) {}^{2} = {a}^{2} + 2ab + {b}^{2} [/tex]
Here, a is sin x
b is cos x
So
[tex]( \sin(x) + \cos(x) ) {}^{2} = \sin {}^{2} (x) + 2 \sin(x) \cos(x) + \cos {}^{2} (x) [/tex]
We know
[tex] \sin {}^{2} (x) + \cos {}^{2} (x) = 1[/tex]
So we know have
[tex]1 + 2 \sin(x) \cos(x) [/tex]
[tex] \sin(2x) = 2 \sin(x) \cos(x) [/tex]
So our final answer is
[tex]1 + \sin(2x) [/tex]
The sales decay for a product is given by
S = 80,000e−0.5x,
where S is the monthly sales and x is the number of months that have passed since the end of a promotional campaign.
(a) What will be the sales 2 months after the end of the campaign? (Round your answer to two decimal places.)
$
(b) How many months after the end of the campaign will sales drop below $1,000, if no new campaign is initiated? (Round up to the nearest whole number.)
months
a) The sales 2 months after the end of the campaign will be $29,600. b) it will take 10 months (round up to the nearest whole number) for sales to drop below $1,000, if no new campaign is initiated.
The sales 2 months after the end of the campaign can be found by plugging x = 2 into the equation:
S = 80,000e-0.5x
S = 80,000e-0.5*2
S = 80,000e-1
S = 80,000 * 0.37
S = 29,600
Therefore, the sales 2 months after the end of the campaign will be $29,600.
We can solve this by setting the equation equal to 1000 and solving for x:
1000 = 80,000e-0.5x
1000/80,000 = e-0.5x
0.0125 = e-0.5x
ln(0.0125) = ln(e-0.5x)
-4.81 = -0.5x
x = 9.62
Therefore, it will take 10 months (round up to the nearest whole number) for sales to drop below $1,000, if no new campaign is initiated.
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PLEASE I REALLY NEED HELP ASAP
[5 points] Each size of tile is named for its area. The smallest tile, called the “unit tile”, has sides that measure exactly 1 unit. Therefore, the area of the unit tile is 1 unit 1 unit=1 unit2. Can you use the unit tile to find the exact area of the other tiles? Explain.
Using the unit tile the area of the three diagrams is - square tile = 1 unit², rectangular tile = x unit², and square tile = x² unit².
What is area?
An object's area is how much space it takes up in two dimensions. It is the measurement of the quantity of unit squares that completely cover the surface of a closed figure.
For the square tile with length and breadth = 1 unit, the area is simply the product of the length and breadth, which is 1 unit × 1 unit = 1 unit².
This is the same as the area of the unit tile, so we don't really need to use it to find the area of this tile.
For the rectangular tile with length = x units and breadth = 1 unit, we can use x unit tiles to cover the length, and 1 unit tiles to cover the breadth.
Therefore, the area of the rectangular tile is x unit × 1 unit = x unit².
For the square tile with length and breadth = x units, we can use x unit tiles to cover the length, and x unit tiles to cover the breadth.
Therefore, the area of the square tile is x unit × x unit = x² unit².
In general, for any tile with length = a units and breadth = b units, the area is given by the product of the length and breadth, which is a unit × b unit = ab unit².
Therefore, we can use the unit tile to find the exact area of any tile, as long as we know its dimensions.
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Consider the line x+4y=4
The slope of a line perpendicular to given line is 4.
What is slope of a line?The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.
The given equation of a line is x+4y=4.
Here, 4y=4-x
y=1-x/4
So, slope m is -1/4
The slope of a line perpendicular to given line is m1=-1/m2
Slope of perpendicular to line is -1/(-1/4)
= 4
Therefore, the slope of a line perpendicular to given line is 4.
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The slope of parallel line is,
m = - 1/4
And, Slope of perpendicular line are,
m = 4
What is Equation of line?The equation of line in point-slope form passing through the points
(x₁ , y₁) and (x₂, y₂) with slope m is defined as;
⇒ y - y₁ = m (x - x₁)
Where, m = (y₂ - y₁) / (x₂ - x₁)
Given that;
The equation of line is,
⇒ x + 4y = 4
Now, We can write as;
⇒ x + 4y = 4
⇒ 4y = - x + 4
⇒ y = - 1/4x + 4
Hence, The slope of the line is,
⇒ y = - 1/4x + 4
⇒ dy/dx = - 1/4
⇒ m = - 1/4
Now, We know that;
The slope of parallel line are equal.
Hence, The slope of parallel line is,
m = - 1/4
And, Product of Slopes of perpendicular lines are - 1.
Hence, Slope of perpendicular line are,
m = - 1/ (- 1/4)
m = 4
Thus, The slope of parallel line is,
m = - 1/4
And, Slope of perpendicular line are,
m = 4
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Find all zeros (real and complex ) of the polynomial x^(4)+2x^(3)+22x^(2)+50x-75=0
The zeros of the polynomial x4 + 2x3 + 22x2 + 50x - 75 = 0 are x = 3, x = -5, x = ±√(37).
To find all the zeros (real and complex) of the polynomial x^(4)+2x^(3)+22x^(2)+50x-75=0, we can use the Rational Root Theorem and synthetic division.
The Rational Root Theorem states that if p/q is a rational root of a polynomial equation, then p must be a factor of the constant term and q must be a factor of the leading coefficient.
The factors of the constant term -75 are: ±1, ±3, ±5, ±15, ±25, ±75
The factors of the leading coefficient 1 are: ±1
Therefore, the possible rational roots of the polynomial are: ±1, ±3, ±5, ±15, ±25, ±75
We can use synthetic division to test each of these possible roots until we find one that is a root. Once we find a root, we can use synthetic division again to divide the polynomial by the factor (x - root) to get a smaller polynomial, and then repeat the process until we have found all the roots.
Using synthetic division, we find that 3 is a root of the polynomial. Dividing the polynomial by (x - 3) gives us a smaller polynomial: x^(3)+5x^(2)+37x+25=0
We can repeat the process with this smaller polynomial to find the remaining roots. Using synthetic division again, we find that -5 is a root of the smaller polynomial. Dividing the smaller polynomial by (x + 5) gives us an even smaller polynomial: x^(2)+37=0
This polynomial has no real roots, but it has two complex roots: x = ±√(-37) = ±√(37)i
So, the complete list of zeros (real and complex) of the original polynomial is: 3, -5, ±√(37).
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When copying segments and angles, which step is the same
And no the answer isn’t draw a ray with one endpoint I get it wrong every time I get the answer
When copying segments and angles, drawing a ray with one endpoint step is the same.
What is a line segment?A line segment is a part of a continuous line, having two fixed ends and a definite length.
When a common part in constructing a line segment and an angle is drawing a ray because a ray is a common part, when we see the figures of a segment and an angle we see a common a ray, so a ray is the common part in both.
Hence, when copying segments and angles, drawing a ray with one endpoint step is the same.
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Which of the following encourages creativity and innovation? A. flexibility B. Realistic expectations C. Persistence D. Organized planning
In respοnse tο the query, we can state that While persistence might be equatiοn crucial in the pursuit οf οriginal ideas, it is insufficient οn its οwn tο fοster creativity.
A. Flexibility fοsters inventiοn and creativity.
What is equatiοn?In a math equatiοn, twο assertiοns are cοnnected by the equals sign (=), which denοtes equivalence. A mathematical assertiοn used in algebraic equatiοns establishes the equivalence οf twο mathematical statements. Fοr instance, in the equatiοn 3x + 5 = 14, the equal sign creates a space between the values 3x + 5 and 14.
Tο cοmprehend the relatiοnship between the twο sentences written οn οppοsing sides οf a letter, utilise a mathematical fοrmula. The lοgο and the specific prοgramme typically cοrrespοnd. An illustratiοn wοuld be 2x - 4 = 2.
Peοple and οrganizatiοns whο are flexible are better equipped tο adjust tο shifting cοnditiοns, explοre new avenues, and adapt. Its adaptability encοurages experimentatiοn and taking chances, which can result in fresh, creative ideas.
Fοr reaching οbjectives and cοmpleting wοrk quickly, realistic expectatiοns and well-οrganized preparatiοn are crucial, but they may nοt always fοster creativity and inventiοn. While persistence might be crucial in the pursuit οf οriginal ideas, it is insufficient οn its οwn tο fοster creativity.
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If a random variable X has exponential distribution with mean 1 then P[X > 2] is
a. 1- e^-2 b. e^2 c. e^-2
d. 1-e²
The correct answer is option a. 1 - e^-2.
To find the probability of a random variable X with exponential distribution, we use the following formula:P[X > x] = e^(-λx)Where λ is the rate parameter and x is the value we are trying to find the probability of.
In this case, we are given that the mean of the distribution is 1, so we can use this information to find the rate parameter:λ = 1/mean = 1/1 = 1Now, we can plug in the values for λ and x into the formula to find the probability:P[X > 2] = e^(-1*2) = e^-2 = 0.1353Therefore, the correct answer is option a. 1 - e^-2.
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Need help multiply fractions and whole numbers
Answer:
Step-by-step explanation:
5. 15 multiplied by 4/5 = 3 multiplied by 4 = 12.
6. 3/11 multiplied by 66 = 3 multiplied by 6 = 18.
write two equivalent rational numbers of 7/-8
Answer:
-14/16
-35/40
Step-by-step explanation:
7/-8 times [tex]\frac{2}{2}[/tex] is -14/16
7/-8 times [tex]\frac{5}{5}[/tex] is -35/40
Answer:
7-8-87-87-87-7-767&7&7
Step-by-step explanation:
そう笑ここに何を書けばいいのかわからないので、ここで私は答えを知っているふりをしています
a discount of $193.16 to its regular price of $828.15
sales price= $
Answer: $634.99
Step-by-step explanation:
Regular price, R - Discount, D = S
subtract discount from regular price
Answer: $634.99
Step-by-step explanation:
$828.15 - 193.16 would equal 634.99
so the discounted price would be $634.99
5. [3 points) Consider the vectors v1 = |2| v2= |3|, v3=|1| . |2| |1| |h| . |-4| |-4| |4| |Determine all values of the parameter h such that the vectors Vi, V2 and v3 are linearly dependent. You should justify your answer.
Given vectors are v1 = |2| v2 = |3| v3 = |1| |2| |1| |h| |-4| |-4| |4| Let's consider the vectors are linearly dependent. If they are linearly dependent, then the rank of the matrix will be less than 3. So, Values of the parameter h are |2| 3 1 |2| 1 h |-4| |-4| 4
For the matrix to have a rank of less than 3, it must satisfy the following condition : |2| 3 1 |2| 1 h |-4| |-4| 4It can be rewritten as follows: |2| 3 1 |2| 1 h |-4| |-4| 4It can be simplified as follows: |2| 3 1 |2| 1 h |-4| |-4| 4. The matrix can be reduced to the following form: |2| 3 1 |2| 1 h |-4| |-4| 4, By subtracting R1 from R2, we get: |-1| 0 |h-1| |-6| |-h| |-4| |-4| 4The determinant of this matrix is: -4h + 4 + 8h = 4h + 4
Now, let's evaluate the matrix's rank. We can observe that the matrix's second row is equal to twice the matrix's first row. So, the rank of the matrix is 2.Since the rank of the matrix is 2, the given vectors are linearly dependent if h = -1.
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Jerry is a judge. He hears
5
55 cases every
2
3
8
2
8
3
2, start fraction, 3, divided by, 8, end fraction hours. Jerry hears cases at a constant rate.
How many cases does he hear per hour?
The number of cases which Judge Jerry hears every hour is 2 2/19
What is a Fraction?An element of a whole or, more broadly, any number of equal pieces, is represented by a fraction.
When used in conversational English, a fraction indicates the number of components of a particular size, as in one-half, eight-fifths, and three-quarters.
How to calculate:
GIven that he tries 5 cases in 2 3/8 hours
The unit rate of cases per hour is calculated by dividing the number of cases by the number of hours.
5/(2 3/8) = 5/(19/8) = 5 * 8/19 = 40/19 = 2 2/19
Therefore, the number of cases which Judge Jerry hears every hour is 2 2/19
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How many one-thirds are in seven-halves
Answer: 14/3 or 3/4 are in 7/2
Step-by-step explanation:
Find the final amount for an investment of $5,000 over 5 years at an annual interest rate of 6% if the interest is compounded quarterly.
The final amount for the investment of $5,000 over 5 years at an annual interest rate of 6% compounded quarterly is $ 6,749.29.
Compound interest:To find the final amount for an investment P over t years at an annual interest rate of r compounded quarterly, we can use the formula for compound interest:
[tex]{\displaystyle A=P\left(1+{\frac {r}{n}}\right)^{nt}}[/tex]
Where:
A = Final amount
P = Principal (initial investment)
r = Annual interest rate (as a decimal)
n = Number of times the interest is compounded per year
t = Time period in years
Here we have
P = $5,000,
r = 6% = 0.06,
n = 4 (since the interest is compounded quarterly), and
t = 5 years.
Using the above formula
[tex]A = 5000(1 + 0.06/4)^{(4*5)[/tex]
[tex]A = 5000(1.015)^{20}[/tex]
[tex]A = 5000(1.34985711)[/tex]
[tex]A = $6,749.29[/tex]
Therefore,
The final amount for the investment of $5,000 over 5 years at an annual interest rate of 6% compounded quarterly is $ 6,749.29.
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Out of 25 26 27 28 29 30 31 and 32 which one is a factor of 27
Answer:
27
Step-by-step explanation:
factors are numbers that divide exactly into a number , leaving no remainder.
then the only factor of 27 from the list is 27
Joey started at point A and walked 20 m
south, 40 m west and a further 50 m south
to arrive at point B. Lana started at point A
and walked in a straight line to point B.
How much further did Joey walk than Lana?
Give your answer in metres (m) to 1 d.p.
A
✓ Scroll down
Joey walked 110 meters from A to B and Lana started at point A and walked in a straight line to point B which is a distance of about 80.6 meters, obtained using Pythagorean Theorem, therefore;
Joey walked about 29.4 meters further than Lana.
What is the Pythagorean Theorem?Pythagorean Theorem states that the square of the length of the hypotenuse side of a right triangle is the sum of the squares of the lengths of the legs of the right triangle.
The distance Joey walked = 20 m + 40 m + 50 m = 110 m
The distance Lana walked can be found as follows;
The similar triangles formed by the path Lana walked and the path Joey walked, indicates that the ratio of the base lengths of the right triangles are;
50/20 = 5/2
The base length of the larger right triangle 5/(2 + 5) × 40 m = 200/7 m
Base length of the smaller right triangle = 2/(5 + 2) × 40 m = 80/7 m
The length of the path Lana walked, l, found using Pythagorean Theorem is therefore;
l = √((20 m)² + ((80/7) m)²) + √((50 m)² + ((200/7) m)²) ≈ 10·√(65) m ≈ 80.6 mThe distance further Joey walked = 110 m - 80.6 m = 29.4 m
Joey
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Determine whether each expression is equivalent or not equivalent to 16^t-0.25?
A) 4^2t/4^0.5
B) 16^t/2^4
C) 16^t/2
The sοlutiοn οf the given prοblem οf expressiοn cοmes οut tο be examples abοve are equal tο 16t - 0.25.
What is the expressiοn?Mathematical prοcesses like multiplying, dividing, adding, and nοw even deleting are required. If they were merged, the fοllοwing claim wοuld be made: An equatiοn, sοme statistics, and a mathematical fοrmula Values, cοmpοnents, and mathematical prοcesses like additiοns, subtractiοns, reversals, algebraic fοrmulas, and divisiοns make up a statement οf truth. Wοrds and phrases can be rated and analyzed.
Here,
We pοssess 16t – 0.25.
A) The fοrmula 42t/40.5 equals (22t)² / (20.5) = 24t / (20.5) = 16t / (20.5)4 = 16t / 4 = 4(16t).
Expressiοn A is nοt equal because 4(16t) is nοt the same as 16t - 0.25.
B) [tex](16)^t/2^4 = (2^4)^t / 2^4[/tex]
=> [tex]2^{4t} / 2^4 = 2^{(4t-4)} = 16^{(t-1)} (t-1).[/tex]
Expressiοn B is not equivalent because 16(t-1) is nοt equal to 16(t - 0.25).
C) [tex]16^t/2 = (2^4)^t / 2[/tex]
[tex]= > 2^{4t} / 2 = 2^{(4t-1)} = 16^t \times 0.5.[/tex]
Expression C is nοt equivalent because 16t * 0.5 does not equal 16t - 0.25.
Hence, As a result, nοne of the examples above are equal to 16t - 0.25.
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Answer:
Step-by-step explanation:
what is the rule multiplying with same sign
Answer:
When you multiply two numbers with the same sign (either both positive or both negative), the rule is that the result is always positive.
For example, if you multiply +2 and +3, the result is +6 because both numbers have the same positive sign. Likewise, if you multiply -4 and -5, the result is +20 because both numbers have the same negative sign.
This rule applies to any two numbers with the same sign, regardless of their values or whether they are whole numbers, fractions, or decimals.
b
P
H
40 50 54
70
84
87 90
Referring to the figure above, which numbers are considered
possible outliers?
●
31
40, 84
31, 87, 90
84, 87, 90
31, 40, 50
.
Based on the box-and-whisker plot, the numbers that are considered possible outliers include the following: B. 31, 87, 90.
What is an outlier?In Mathematics, an outlier is also referred to as anomalous data and it can be defined as a numerical value that is either unusually too small or large (big) in comparison with the overall pattern of the numerical values contained in a data set.
What is a box-and-whisker plot?In Mathematics, a box plot is sometimes referred to as box-and-whisker plot and it can be defined as a type of chart that can be used to graphically or visually represent the five-number summary of a data set with respect to locality, skewness, and spread.
By critically observing the box-and-whisker plots or box plot, we can logically deduce that 31, 87, and 90 are possible outliers.
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The value of the series 4+9+14+dots +(5n-1) by using the summation notation equals
The value of the series 4 + 9 + 14 + ... + (5n-1) using the summation notation is (5n^2 + 3n)/2.
The value of the series 4 + 9 + 14 + ... + (5n-1) can be found using the summation notation as follows:
Identify the first term, common difference, and number of terms in the series. The first term is 4, the common difference is 5, and the number of terms is n.
Use the summation notation to write the series as ∑_{i=1}^{n} (5i-1).
Use the formula for the sum of an arithmetic series to find the value of the series. The formula is S_n = n/2 (a_1 + a_n), where S_n is the sum of the series, n is the number of terms, a_1 is the first term, and a_n is the last term.
Substitute the values of n, a_1, and a_n into the formula and simplify. S_n = n/2 (4 + (5n-1)) = n/2 (5n+3) = (5n^2 + 3n)/2.
The value of the series is (5n^2 + 3n)/2.
Therefore, the value of the series 4 + 9 + 14 + ... + (5n-1) using the summation notation is (5n^2 + 3n)/2.
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Connor and Helen are playing a matching game to practice working with the laws of exponents. Connor's card has 18 . Helen has 3 cards to choose from. One card has 2−3 . Another card has (12)−2 . The final card has (22)−4 . Complete the statements below for each of Helen's cards.
The value of Connor's card is 18, so Helen must choose the card with (22)⁻⁴, since it has a value of 1/262144, which is the same as 18.
What is matching game?Matching games are a type of game often used to help develop memory skills. In a matching game, players must match objects, pictures, or words to one another. This can involve matching a picture to a word, or a word to a definition. Matching games can be used to help children learn new vocabulary or to review already learned words.
The card with 2⁻³ has a base of 2 and an exponent of -3. This means that the value of the card is 1/8, which is the same as 2⁻³.
The card with (12)⁻² has a base of 12 and an exponent of -2. This means that the value of the card is 1/144, which is the same as 12⁻².
The card with (22)⁻⁴ has a base of 22 and an exponent of -4. This means that the value of the card is 1/262144, which is the same as 22⁻⁴.
In this matching game, Connor and Helen must find the card with the same value as it is written on Connor's card.
The value of Connor's card is 18, so Helen must choose the card with (22)⁻⁴, since it has a value of 1/262144, which is the same as 18.
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Let sine of theta equals the quantity 2 times radical 3 end quantity over 5 and pi over 2 is less than theta is less than pi period
Part A: Determine the exact value of cos 2θ.
Part B: Determine the exact value of sine of the quantity theta over 2 end quantity period
The exact value οf cοs(2θ) is 27/25.
The exact value οf sin(θ/2) is -√[(5 + √13) / 10].
What is Trigοnοmetry?Trigοnοmetry is a branch οf mathematics that deals with the relatiοnships between the sides and angles οf triangles, and the trigοnοmetric functiοns that describe thοse relatiοnships.
Given: sin(θ) = (2√3) / 5, and π/2 < θ < π.
Part A:
cοs(2θ) = 2cοs²(θ) - 1
We can find cοs(θ) using the Pythagοrean identity:
cοs²(θ) + sin²(θ) = 1
cοs²(θ) = 1 - sin²(θ)
cοs(θ) = ±√(1 - sin²(θ))
Since π/2 < θ < π, sin(θ) is pοsitive and cοs(θ) is negative in the secοnd quadrant. Therefοre, we have:
cοs(θ) = -√(1 - (2√3/5)²) = -√(1 - 12/25) = -√13/5
Substituting intο the fοrmula fοr cοs(2θ), we get:
cοs(2θ) = 2cοs²(θ) - 1 = 2(-√13/5)² - 1 = 52/25 - 1 = 27/25
Therefοre, the exact value οf cοs(2θ) is 27/25.
Part B:
We can use the half-angle fοrmula fοr sine tο find sin(θ/2):
sin(θ/2) = ±√[(1 - cοs(θ)) / 2]
Since π/2 < θ < π, sin(θ/2) is negative in the secοnd quadrant. Therefοre, we have:
sin(θ/2) = -√[(1 - cοs(θ)) / 2] = -√[(1 - (-√13/5)) / 2] = -√[(5 + √13) / 10]
Therefοre, the exact value οf sin(θ/2) is -√[(5 + √13) / 10].
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When solving this equation for x, what would you do first? (3(x+4))/(5)=6 Divide both sides by 5 . Multiply both sides by 5 .
The first step would be to multiply both sides by 5 when solving the equation (3(x+4))/(5)=6.
This will allow us to eliminate the denominator on the left side of the equation and simplify the equation.After multiplying both sides by 5, the equation becomes: 3(x+4) = 30
Next, we can distribute the 3 on the left side of the equation: 3x + 12 = 30
Then, we can subtract 12 from both sides of the equation: 3x = 18
Finally, we can divide both sides by 3 to solve for x: x = 6
Therefore, the solution to the equation is x=6.
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Feb 23, 9:03:56 PM Watch help video Express (x+5)^(2) as a trinomial in standard form.
The trinomial in standard form is x^2 + 10x + 25.
To express (x+5)^(2) as a trinomial in standard form, we need to expand the expression using the distributive property.
First, we will distribute the first term, x, to each term inside the parentheses:
(x+5)(x+5) = x(x) + x(5) + 5(x) + 5(5)
Next, we will simplify the terms:
= x^2 + 5x + 5x + 25
Finally, we will combine like terms to get the trinomial in standard form:
= x^2 + 10x + 25
Therefore, the trinomial in standard form is x^2 + 10x + 25.
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An airplane is flying on a compass heading (bearing) of 170 deg at 460 mph. A wind is blowing with the bearing 200 deg at 80 mph.
a. Find the component form of the velocity of the airplane. b. Find the actual ground speed and direction of the airplane
Answer:
a. v = 460 < cos 170 sin 170 >=<-453.01,79.88>
b. Find the wind vector w = 80 < cos 200 sin 200 >=<-75.18. - 27.36 >
Velocity vector = v + w <= - 528.19 . 52.52>
Actual speed |v+w| sqrt((- 528.19) ^ 2 + (52.52) ^ 2) approx530.79 mph
Actual direction: Theta = 180 deg + arctan(- 528.19/52.52) = 95.68 deg
For the airplane, we have v = -453.01, 79.88. Similarly, for the wind, we have w = -75.18, -27.36.
The actual speed would be 530.79 mph.
The actual ground speed and direction of the airplane are 530.79 mph and 174.32 deg, respectively.
The component form of the velocity of the airplane can be found by using the formula:
v = r < cos theta, sin theta >, where r is the speed and theta is the bearing. For the airplane, we have v = 460 < cos 170, sin 170 > = <-453.01, 79.88>. Similarly, for the wind, we have w = 80 < cos 200, sin 200 > = <-75.18, -27.36>.
To find the actual ground speed and direction of the airplane, we need to add the velocity vector of the airplane and the wind vector. This gives us the velocity vector = v + w = <-453.01, 79.88> + <-75.18, -27.36> = <-528.19, 52.52>.
The actual speed can be found by taking the magnitude of the velocity vector, which is given by |v+w| = sqrt((-528.19)^2 + (52.52)^2) = 530.79 mph.
The actual direction can be found by using the formula theta = arctan(y/x), where x and y are the x and y components of the velocity vector. In this case, we have theta = arctan(52.52/-528.19) = -5.68 deg. However, since the velocity vector is in the third quadrant, we need to add 180 deg to get the actual direction, which is 180 + (-5.68) = 174.32 deg.
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Samantha mixes some amount of 25% sugar syrup with x grams of 10% sugar syrup. The result is 120 grams of 15% sugar syrup.
How much pure sugar is in 120 grams of 15% syrup?
HELP ASAP PLEASSEEE !!
Answer:
Let's start by using the formula for mixing two solutions to set up an equation. The formula is:
(concentration of solution 1) x (volume of solution 1) + (concentration of solution 2) x (volume of solution 2) = (concentration of resulting solution) x (total volume of resulting solution)
In this problem, solution 1 is the 25% sugar syrup, solution 2 is the 10% sugar syrup, and the resulting solution is 15% sugar syrup. We are also given that the total volume of the resulting solution is 120 grams. Let's use x to represent the volume of the 10% sugar syrup that is mixed with the 25% sugar syrup. Then we can write:
(0.25)(120-x) + (0.10)(x) = (0.15)(120)
Simplifying this equation, we get:
30 - 0.25x + 0.10x = 18
0.15x = 12
x = 80
Therefore, we need to mix 80 grams of the 10% sugar syrup with 40 grams of the 25% sugar syrup to get 120 grams of 15% sugar syrup.
To find the amount of pure sugar in the 120 grams of 15% syrup, we can use the fact that the concentration of pure sugar in the resulting solution is 15%. That means that 15% of the 120 grams is pure sugar. We can find this by multiplying 120 by 0.15:
120 x 0.15 = 18
So there are 18 grams of pure sugar in the 120 grams of 15% syrup.