Write an equivalent expression in simplest form for each given expression. Then, choose the answer that explains why the expressions are equivalent.
(A) 2y + 4(y + 1) = y +
First use the Property to simplify the expression in parentheses, then
combine like terms.
(B) 9y + 6 + 2(5 + y) = y +
After applying the Distributive Property, use the Properties to
combine like terms.
The equivalent expressions are 6y + 4 and 11y + 16
How to determine the equivalent expressions?Expression (A)
We have:
2y + 4(y + 1)
Open the bracket
2y + 4y + 4
Evaluate the like terms
6y + 4
Expression (B)
We have:
9y + 6 + 2(5 + y)
Open the bracket
9y + 6 + 10 + 2y
Evaluate the like terms
11y + 16
Hence, the equivalent expressions are 6y + 4 and 11y + 16
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Kite A B C D is shown. Lines are drawn from point A to point C and from point B to point D and intersect.
Figure ABCD is a kite. The area of ABCD is 48 square units. The length of line segment BD is 8 units.
What is the length of AC?
5 units
6 units
10 units
12 units
The length of AC is 12 units . Option D
How to determine the length
The formula for area of a kite is given thus;
Area = [tex]\frac{p * q}{2}[/tex]
Where p and q are the lengths
Area of kite = 48 square units
BD, q = 8 units
AC , p = unknown
Substitute the values
[tex]48 = \frac{AC * 8}{2}[/tex]
Cross multiply
[tex]AC * 8 = 48 * 2[/tex]
[tex]8AC = 96[/tex]
[tex]AC = \frac{96}{8}[/tex]
[tex]AC = 12[/tex] units
Thus, the length of AC is 12 units . Option D
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Answer:option 4
Step-by-step explanation:
Find an equation for the plane that contains the line
v=(-3,4,5)+t(3,2,4)
and is perpendicular to the plane
2x+y-3z+4=0
(Use symbolic notation and fractions where needed.)
The equation for the plane that contains the line and is perpendicular to the plane 2 · x + y - 3 · z = - 4 is - 10 · x + 17 · y - z = 76.
How to find the equation of a plane that contains a line and is perpendicular to another plane
From the equation of the plane 2 · x + y - 3 · z = - 4 we know that its normal vector is (2, 1, - 3). Hence, we have found two direction vectors and already know a point, which are part of this parametric equation:
(x, y, z) = (- 3, 4, 5) + t · (3, 2, 4) + u · (2, 1, - 3) (1)
The normal vector of the plane is equal to the cross product of the direction vectors of (1):
[tex]\vec n = (3, 2, 4) \,\times\,(2, 1, -3)[/tex]
[tex]\vec n = \left|\begin{array}{ccc}\hat{i}&\hat{j}&\hat{k}\\3&2&4\\2&1&-3\end{array}\right|[/tex]
[tex]\vec n = (-6-4,8+9, 3-4)[/tex]
[tex]\vec n = (-10, 17, -1)[/tex]
Since the plane contains the point (x, y, z) = (- 3, 4, 5), then we find that the independent constant of the equation of the plane is:
- 10 · x + 17 · y - z = k
- 10 · (- 3) + 17 · 4 - 5 = k
30 + 51 - 5 = k
k = 76
The equation for the plane that contains the line and is perpendicular to the plane 2 · x + y - 3 · z = - 4 is - 10 · x + 17 · y - z = 76.
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A bag contains 3 red balls, 4 green balls, and 2 yellow balls. Deborah reaches into the bag and pulls out the following 4 balls: red, red, green, yellow Deborah reaches into the bag to pick out another ball. What's the probability that the ball she picks is green?
The probability that the ball Deborah picks is green is; 3/5.
What is the probability that the ball picked last is green?Since the bag initially contains; 3 red balls, 4 green balls, and 2 yellow balls, after Deborah pulls out the 4 balls: red, red, green, yellow.
The balls remaining are; 1 red ball, 3 green and 1 yellow.
Hence, the probability that the final ball she picks is green is; 3/(3+1+1) = 3/5.
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Tim wants to build a rectangular fence around his yard. He has 42 feet of fencing. If he wants the length to be twice the width. What is the largest possible length
Answer: The largest he can possibly build is x = 14
Step-by-step explanation:
You have nine line segments with the lengths of 1, 2, 3, 4, 5, 6, 7, 8, and 9, respectively.
How many ways are there to form a square by connecting the ends of some of these
line segments? No overlapping of the line segments is allowed.
The number of ways to forma a square by connecting the ends of some of the line segments is 3, 024 ways
What is permutation?From the given information, we should use the formula for permutation without repetition.
The formula is given as;
Permutation = [tex]\frac{n!}{n -r !}[/tex]
Where n = number of set = 9
r = 4, this is so because, the sides of a square a four
Permutation = [tex]\frac{9!}{9 - 4!}[/tex]
Permutation = [tex]\frac{9!}{5!}[/tex]
Permutation = [tex]\frac{362, 880}{120}[/tex]
Permutation = 3, 024 ways
Thus, the number of ways to forma a square by connecting the ends of some of the line segments is 3, 024 ways
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Which equation is represented by the graph below?
A. Y= 1/8e^x
B. Y= 1/2e^x
c. Y=2e^x
D. Y= 8e^x
#49
Answer:
B
Step-by-step explanation:
When x = 1, it appears y is in between 1 and 2.
The only choice that satisfies this is B.
Answer:
The equation that represents the graph is y=2e^x
Step-by-step explanation:
plug in geogebra and confirm:D
Determine the equation (picture).
The line perpendicular to the equation and has the same y-intercept with another is y = - 4 / 3 x + 4.
How to find equation of a line?The equation of a line can be describe as follows:
y = mx + b
where
m = slopeb = y-interceptTherefore, lines that a perpendicular follows the rule below:
m₁m₂ = -1
Hence,
y + 6 = 3 / 4 (x - 2)
y + 6 = 3 / 4 x - 6 / 4
y = 3 / 4 x - 6 / 4 - 6
y = 3 / 4 x - 30 / 4
y = 3 / 4 x - 15 / 2
Hence,
3 / 4 m₂ = -1
m₂ = - 4 / 3
Therefore,
slope of the line is - 4 / 3
Let's find the y-intercept of the second line
4x + 5y -20 = 0
5y = -4x + 20
y = -4 / 5 x + 4
y-intercept = 4
Therefore, the line perpendicular to the equation and has the same y-intercept with another is y = - 4 / 3 x + 4.
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An issue of Taunton's Fine Woodworking included plans for a hall stand. The total height of the stand is 5912 59 1 2 inches. If the base is 25716 25 7 16 inches, how tall is the upper portion of the stand?
The length if the upper portion of the stand is 33. 56 inches
How to determine the lengthThe formula for the finding the length is given thus;
Total height = length of upper portion + length of base
Where
Length of base = 257/16
Total height = 591/2
Length of upper portion = total height - length of base
= 118/2 - 407/16
= 59 - 25. 44
= 33. 56 inches
Thus, the length if the upper portion of the stand is 33. 56 inches
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Use properties of operations to find the quotient. -10.2/6
A. 1.7 B. -1.7 C. 4.2 D. -4.2
Answer:
B, -1.7
Step-by-step explanation:
-10.2 divided by 6 is -1.7.
I hope this helps!
Practice 3
Solving a Special Solution inequality
17<3h+2<2
h>______
h <______
Answer:
h > 5
h < 0
Step-by-step explanation:
We suppose that a "Special Solution" inequality is one that is technically incorrect, but is written using a compact "shorthand" form.
As written, the inequality claims that 17 < 2, which is false. There can be no value of the variable that will make this true. We presume the intended meaning is ...
17 < 3h +2 or 3h +2 < 2
Special solutionSubtracting 2 from the inequality gives ...
15 < 3h < 0
Dividing by 3 gives ...
5 < h < 0
There is no value of h that is both greater than 5 and less than 0, so we presume this means the OR (union) of the solution sets ...
h > 5
h < 0
find the maximum value of the function 7-2x-3x^2 and the equation of the axis of the curve
The maximum value of the function 7 - 2x - 3x² exists 22/3.
How to estimate the maximum value of the function 7 - 2x - 3x²?Given: 7 - 2x - 3x²
We must begin by estimating the first derivative:
Differentiating the equation above, we have:
dy/dx = 0 - 2 - 6x = - 2 - 6x
We know that dy/dx = 0 at maxima and minima
Therefore, we contain, - 2 - 6x = 0
6x = - 2
x = - 1/3
Substituting x = - 3 back into the equation of the curve yields the following result:
y = 7 - 2(-1/3) - 3(-1/3)² = 22/3
y = 22/3
Therefore, 22/3 exists the maximum value of y.
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A cone-shaped paper water cup has a height of 12 cm and a radius of 6 cm. If the cup is filled with water to one-third its height, what portion of the volume of the cup is filled with water?
Given the graph of g(x), describes the transformation of the parent function f(x)=2^x
Answer:
The transformation being described is from g(x)=x2 g ( x ) = x 2 to f(x)=x2 f ( x ) = x 2 . The horizontal shift depends on the value of h . The horizontal shift is described as: f(x)=f(x+h) f ( x ) = f ( x + h ) - The graph is shifted to the left h units
`H_0: p = 0.63`
`H_1: p > 0.63`
Your sample consists of 150 subjects, with 96 successes. Calculate the test statistic, rounded to 2 decimal places
`z=`
The test statistic, rounded to 2 decimal places is equal to
How to calculate value of the test statistic?For this sample, the hypothesis is given by:
H₀: μ₁ ≤ μ₂
H₁: μ₁ > μ₂
Assuming this sample has a normal distribution, we would use a pooled z-test to determine the value of the test statistic:
Substituting the given parameters into the formula, we have;
[tex]z = \frac{\frac{96}{150} \;-\;0.63}{\sqrt{0.63\; +\; \frac{1\;-\;0.63}{150}} }\\\\z = \frac{0.64 \;-\;0.63}{\sqrt{0.63\; +\; \frac{0.37}{150}}}\\\\z = \frac{0.01}{\sqrt{0.63\; +\; 0.0025}}\\\\z = \frac{0.01}{\sqrt{0.6325}}\\\\[/tex]
z = 0.01/0.7953
z = 0.013.
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Determine the area under the standard normal curve that lies to the left of (a) Z=081. (b) Z=0.43. (c) Z= -0.33. and (d) Z= 1.04.
Using the normal distribution, the areas to the left are given as follows:
a) 0.7910.
b) 0.6664.
c) 0.3707.
d) 0.8508.
Normal Probability DistributionThe z-score of a measure X of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex] is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The z-score measures how many standard deviations the measure is above or below the mean. Looking at the z-score table, the p-value associated with this z-score is found, which is the percentile of X, and is also the area to the left of Z.Hence:
The area to the left of Z = 0.81 is of 0.7910.The area to the left of Z = 0.43 is of 0.6664.The area to the left of Z = -0.33 is of 0.3707.The area to the left of Z = 1.04 is of 0.8508.More can be learned about the normal distribution at https://brainly.com/question/4079902
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A carpenter built a square table with side length x. Next, he will build a rectangular table by tripling one side and halving the other. The area of the rectangular table is represented by the expression (3 x)(one-half x).
A square with sides x. A rectangle with side one-half x and side 3 x.
What is the simplified expression for the area of the rectangular table?
Three-halves x squared
Three-halves x
Five-halves x squared
Five-halves x
The area of a rectangle is given by its length multiplied by its width. So that the expression that represents the area of the rectangle in the given question is [tex]\frac{3}{2}[/tex] [tex]x^{2}[/tex] squared unit.
A rectangle is a figure that has four straight sides, in which opposite sides are equal.
The area of a rectangle can be determined by the expression;
Area = length x width
Thus considering the given question, the length of the rectangle is tribble to that of the square. And the width of the rectangle is half of that of the square.
So that;
length = 3x
width = [tex]\frac{x}{2}[/tex]
Thus,
Area = length x width
= 3x ([tex]\frac{x}{2}[/tex])
= [tex]\frac{3}{2}[/tex] [tex]x^{2}[/tex]
Therefore, the area of the rectangle is [tex]\frac{3}{2}[/tex] [tex]x^{2}[/tex] squared unit.
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Find the shortest distance from A to B in the diagram below. A. 505‾‾‾‾√ m B. 17 m C. 329‾‾‾√ m D. 10 m
The shortest distance from A to B in the diagram is 17 m.
How to find the shortest distance?The shortest distance from D to B can be found as follows:
using Pythagoras theorem,
c² = a² + b²
where
c is the hypotenusea and b are the other legsTherefore,
DB² = 15² + 8²
DB² = 225 + 64
DB² = 289
DB = √289
DB = 17 m
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Answer:
B, 17 m
Step-by-step explanation:
Founders 23
Construct a 96% confidence interval if a sampling distribution has a mean of 20, a standard deviation of 5, and a size of 100.
Using the t-distribution, the 96% confidence interval is given as follows:
(18.96, 21.04).
What is a t-distribution confidence interval?The confidence interval is:
[tex]\overline{x} \pm t\frac{s}{\sqrt{n}}[/tex]
In which:
[tex]\overline{x}[/tex] is the sample mean.t is the critical value.n is the sample size.s is the standard deviation for the sample.The critical value, using a t-distribution calculator, for a two-tailed 96% confidence interval, with 100 - 1 = 99 df, is t = 2.0812.
The parameters are given as follows:
[tex]\overline{x} = 20, s = 5, n = 100[/tex]
Hence the bounds of the interval are:
[tex]\overline{x} - t\frac{s}{\sqrt{n}} = 20 - 2.0812\frac{5}{\sqrt{100}} = 18.96[/tex]
[tex]\overline{x} + t\frac{s}{\sqrt{n}} = 20 + 2.0812\frac{5}{\sqrt{100}} = 21.04[/tex]
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If a person rows to his favorite fishing spot 21 miles downstream in the same amount of time that he rows 7 miles
upstream and if the current is 7 mph, find how long it takes him to cover 28 miles.
28 miles upstream requires 28/7 = 4 hours.
28 miles downstream requires 28/21 = 4/3 hours, or 1 hour and 20 minutes.
How long does it take him to cover 28 miles?Apply the distance formula:
d = rt
where
d is distance
r is rate or speed
t is time
21 = t*(r+7) = rt+ 7t
7 = t *(r-7) = rt - 7t
28 = 2rt
14 = rt
21 = rt + 7t
21 = 14 + 7t
7 = 7t
t=1
21 = (r+7)*1 = r+7
r = 14
The speed of the boat is 14 mph. With the current (downstream), the speed is 14+7 = 21 mph, so 21 miles are traveled in 1 hour. Against the current (upstream), the speed is 14-7 = 7 mph, which means 7 miles are traveled in the same 1-hour time frame.
28 miles downstream requires 28/21 = 4/3 hours, or 1 hour and 20 minutes.
28 miles upstream requires 28/7 = 4 hours.
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average person burns approximately 221 calories per half-hour while bicycling. If a person does 2 hours of bicycling per day then how many calories will they burn in a week if they work out 5 days a week?
Find the vertex, focus and diretrix of x^2+4x-12y=-64
It needs to be in (x-h)^2=4p(y-k) form
Vertex of the equation is (-2, 5)
Focus of the equation is (-2, 8)
Directrix of the equation is 2
What is a Quadratic curve?
A quadratic curve is a parabolic curve is a graph of the points which define a quadratic function. It shows how a function behaves in the cartesian plane.
Quadratic curve may bend upwards, or downwards depending on the gradient of the curve.
Analysis:
setting the equation in the form, [tex]x^{2}[/tex] +4x -12y = -64 in the form [tex](x-h)^{2}[/tex] = 4p(y-k)
Firstly we take -12y to the right hand side
[tex]x^{2}[/tex] +4x = 12y - 64
Making the left hand side a perfect square expression, we square both sides with -2
[tex]x^{2}[/tex] +4x + [tex](2)^{2}[/tex] = 12y -64 + [tex](2)^{2}[/tex]
[tex](x+2)^{2}[/tex] = 12y -60
[tex](x+2)^{2}[/tex] = 12(y - 5)
comparing with the other equation
-h = , h = -2, -k = - 5 k = 5, 4p = 12, p = 3
Vertex is (h, k) = (-2, 5)
Focus is (h, k+p) = (-2, 5+3) = (-2, 8)
Directrix is y = k-p = 5 - 3 = 2
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Claire has 18 nail polish bottles while Becky has 2/3 of that amount. How many bottles of nail polish does Becky have?
Answer:
15/16
Step-by-step explanation:
because 2/3have Becky
The princess of the Kingdom of Abundance accidentally drops her magic salamander's feeding bowl into a fire. She orders a new feeding bowl to be made out of Dragon Alloy #18, which is 86% magic steel and the rest titanium. The
dwarfs have a lot of Dragon Alloy #33, which is 28% titanium. How much of
that alloy and how much magic steel should they combine to make 700 grams of Dragon alloy #18?
350 grams of magic steel and 350 grams of dragon alloy #33 is needed to make 700 grams of Dragon alloy #18.
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Let x represent the amount of magic steel and y represent the amount of alloy #33, hence:
(0 * x) + (y * 0.28) = 700(1 - 0.86) (1)
Also:
x + (1 - 0.28)y = 0.86(700) (2)
From both equations:
x = 350, y = 350
350 grams of magic steel and 350 grams of dragon alloy #33 is needed to make 700 grams of Dragon alloy #18.
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f is a trigonometric function of the form f(x)=asin(bx+c)+d. The function intersects its midline at (π,−4) and has a minimum point at ( π/4 , − 5.5 ). Find a formula for f(x). Give an exact expression.
The given point on the midline of (π, -4), and the minimum point of (π/4, -5.5), give the exact expression for f(x) as; f(x) = 1.5•sin((2/3)•(x - π)) - 4.
Which method can be used to find the trigonometric function?The amplitude, a = The difference between the y-values at the minimum point and the midline
Therefore;
a = -4 - (-5.5) = 1.5
d = The difference between the y-values of the midline and the x-axis
Therefore;
d = -4 - 0 = -4
The given function is presented as follows;
f(x) = a•sin(b•x + c) + d
Which gives;
-4 = 1.5•sin(b•π + c) - 4
sin(b•π + c) = 0
(b•π + c) = 0
-c/b = π
c = -b•π
Therefore;
-5.5 = 1.5•sin(b•π/4 + c) - 4
-1.5 = 1.5•sin(b•π/4 + c)
-1 = sin(b•π/4 - b•π) = sin(-3•b•π/4)
-3•b•π/4 = arcsine (-1) = -π/2
-3•b•π/4 = -π/2
3•b/4 = 1/2
b = 2/3
c = -(2/3)•π
Therefore, f(x) = a•sin(b•x + c) + d, gives;
f(x) = 1.5•sin((2/3)•x - (2/3)•π) - 4
By simplification, the exact expression for f(x) is therefore;
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I need this answered please and thank you! ASAP
Answer:
Greetings!
The answer for the first figure is attached but the second diagram is not clear.
Also use the image i attached horizontally.
and also it TD subject not Mathematics.
Kaitlin,Scott and bill served a total of 81 orders on Monday .Kaitlin served 9 fewest orders than Scott .bill served 3 times as many orders as Scott . How many orders did they each serve?
Answer: Scott - 18, Kaitlin - 9, Bill - 54
Step-by-step explanation:
Let's view this as units. There are a total of 81 units. If we solve for Scott's orders:
Kaitlin served 9 fewer orders than Scott. We add 9 to 81 --> 9 + 81 = 90.
Now if we view Scott's orders as one unit:
Scott - 1 unit
Kaitlin (after adding 9) - 1 unit
Bill - 3 units (he served 3 times as many orders as Scott)
In total there are 3 + 1 + 1 = 5 units.
90 / 5 = 18.
Therefore, Scott served 18 orders.
18 - 9 = 9.
Kaitlin served 9 orders.
18 x 3 = 54.
Bill served 54 orders.
And to check our answer, 54 + 18 + 9 is 81.
Would the horizontal translations shift the square root right or left? Sort each into the appropriate category.
f(x)=√x+0.8
f(x)=√x-4.7
f(x)=√x-25
f(x)=√x+6
f(x)=√x-11
Answer:
Step-by-step explanation:
Shift right
f(x)=√x-4.7
f(x)=√x-25
f(x)=√x-11
Shift left
f(x)=√x+0.8
f(x)=√x+6
Find the sum of the arithmetic series given a1=45, an=85, and n=. 5
The sum of the arithmetic series is 325
How to determine the sum?The given parameters are
a1 = 45
an = 85
n = 5
The formula for the sum of arithmetic series is:
[tex]S_n = \frac n2 * (a_1 + a_n)[/tex]
Substitute the values of a1, an and n in the above equation
[tex]S_5 = \frac 52 * (45 + 85)[/tex]
Add 45 and 85
[tex]S_5 = \frac 52 * 130[/tex]
Divide 130 by 2
[tex]S_5 = 5 * 65[/tex]
Multiply
[tex]S_5 = 325[/tex]
Hence, the sum of the arithmetic series is 325
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A dairy farmer wants to mix 75% protein supplement and 50% protein ration to make 1400 pounds of a high grade 70% protein ration. How many pounds of each should he use
The dairy farmer would mix 1120 pounds of 75% protein supplement and 280 pounds of 50% protein ration to make 1400 pounds of a high grade 70% protein ration.
What is an equation?An equation is an expression that shows the relationship between two or more variables and numbers.
Let x represent the amount of 75% protein supplement and y represent the amount of 50% protein, hence:
x + y = 1400 (1)
Also:
0.75x + 0.5y = 0.7(1400) (2)
Hence:
x = 1120, y = 280
The dairy farmer would mix 1120 pounds of 75% protein supplement and 280 pounds of 50% protein ration to make 1400 pounds of a high grade 70% protein ration.
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