Answer:
Step-by-step explanation:
i sorry dont know
Find the measure of A and B.
The measure of sides a and b are 10 and √116 respectively.
What are similar triangles?Similar triangles are triangles that have the same shape, but their sizes may vary. The ratio of corresponding sides of Similar triangles are equal.
This means for two triangles to be similar the corresponding angles of the triangles must be equal.
Therefore;
b/29 = 4/b
b² = 29 × 4
b² = 116
b = √116
using Pythagorean theorem,
b² = a²+4²
116 = a² +16
a² = 116-16
a² = 100
a = 10
therefore the measure of both sides a and b are 10 and √116 respectively.
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Solve for x.
A. 2ft
B. 3ft
C. 4ft
D. 6ft
[tex]solution= \frac{x}{3}=\frac{6}{9}- x=2\\ answer:2ft[/tex]Answer:
Step-by-step explanation:
Katie's middle school graduation party will be at the skating rink. The cost for each person is $15. The cost of the room for the party's $80 katies parents have set a budget of $470.
26 kids can come on the budget of 470.
first subtract the 80 for the room then divided the rest by 15
The Question is inside those screenshots
Using the 68-95-99.7 rule
1. The approximate percentage of the Chipotle meals that have between 466 and 1670 calories is 95 percent
2. The middle 99.7% of Chipotle meals have between 165 calories and 1971 calories.
3. The approximate percentage of Chipotle meals with calories between 767 and 1369 is 68%.
How do we calculate the approximate percentage or range using the 68-95-99.7 rule?
The 68-95-99.7 rule states that 68% of the data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3.
1. Within 1 standard deviation, we cannot arrive at 466 and 1670. for example 466 is (1068 - 301) = 767
but wth 2 standard deviations, we can
466 is (1068 - 2 × 301) = 466
1670 is (1068 + 2 × 301) = 1670
2. For 99.7% , we kmow it fall with 3 standard deviation⇒(µ - 3σ) calories and (µ + 3σ) calories.
(1068 - 3 × 301) = 165 calories
(1068 + 3 × 301) = 1971 calories
3. Since we know 95 and 99.7% range, it most likely falls within 68%
(1068 - 301) = 767 and (1068 + 301) = 1369
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U6 Review HW: Practice with Parabolas
1. What are the coordinates of the vertex?
2. What are the coordinates of the focus?
3. What is the equation of the directrix?
a. y = 0
b. y = 1
c. y = 2
d. y = 4
4. Write the equation of the parabola.
2
5. Is the point (2, 4) the same distance from (0, 4) and the line y = 2? Explain why or
why not.
6. Is the point (4, 6) on the parabola? Explain why or why not.
X
Answer:
the vertex is the highest or lowest point on theparabola. (0,3)
The focus is (0,4)
The directrix is the fixed line. Equation y=2
by definition, all points on the parabola are equidistant from a fixed point (focus) and fixed line (directrix)
Step-by-step explanation:
Which of the following is the distance between the two points shown?
A graph with the x-axis starting at negative 4, with tick marks every one-half unit up to 4. The y-axis starts at negative 4, with tick marks every one-half unit up to 4. A point is plotted at negative 3, 0 and at 0.5, 0.
2.5 units
3.5 units
−3.5 units
−2.5 units
The correct answer for the distance between two points is 3.5 units.
To find the distance between two points on a coordinate plane, we can use the distance formula:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
The two points given are (-3,0) and (0.5,0). Plugging these values into the formula, we get:
distance = sqrt((0.5 - (-3))^2 + (0 - 0)^2)
Simplifying the expression inside the square root:
distance = sqrt(3.5^2)
distance = sqrt(12.25)
distance = 3.5
Therefore, the distance between the two points is 3.5 units.
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Suppose the radius of a cylinder changes, but its volume stays the same. How must the height of the cylinder change?
1. If the radius increases then height must increase
2. If radius decreases then height must decrease
3. The height doesn’t change
4. If the radius increases, height decreases
If the radius increases, height decreases, Therefore option no 4 is correct
If the radius of a cylinder changes, however its volume remains the same, then the height of the cylinder have to change.
Mainly, if the radius increases, then the height must decrease & if the radius decreases then the height must to increase. this is because the volume of a cylinder is given by means of the formulation:
[tex]V = \pi r^2h[/tex]
In which V is the volume, r is the radius &h is the height. If we keep the volume constant, and increase the radius, then we ought to lower the height in order to make amends for the increased extent.
Similarly, if we decrease the radius, then we must increase the height to catch up on the reduced volume.
Therefore, the appropriate answer is:
If the radius increases, height decreases
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Sam bought a dog for £110. His nan gave him 30% of the money. How much did he have to pay himself?
Answer:
£77
Step-by-step explanation:
If his nan gave him 30%, he would have to pay the remaining 70%.
70% of 110:
10%=£11
so, 11x7= 77
The royal fruit company produces two types of fruit drinks. The first type is 20% pure fruit juice, and the second type is 70% pure fruit juice. The company is attempting to produce a fruit drink that contains 40% pure fruit juice. How many pints of each of the two existing types of drink must be used to make 150 pints of a mixture that is 40% pure fruit juice?
Hi Caitlin! It is my pleasure to answer this question for you.
Okay, let's denote the number of pints of the first type of fruit drink as x and the number of pints of the second type as y.
From the problem, we know that:
The total number of pints of the mixture is 150: x + y = 150
The first type of drink is 20% pure fruit juice: 0.2x
The second type of drink is 70% pure fruit juice: 0.7y
The resulting mixture is 40% pure fruit juice: 0.4(150) = 60
We can set up two equations based on the amount of pure fruit juice in each type of drink and the resulting mixture:
0.2x + 0.7y = amount of pure fruit juice in the original mixture
0.4(150) = amount of pure fruit juice in the resulting mixture
Substituting the values, we get:
0.2x + 0.7y = 60
x + y = 150
We can solve for x by isolating y in the second equation:
y = 150 - x
Substituting this into the first equation, we get:
0.2x + 0.7(150 - x) = 60
Simplifying and solving for x, we get:
0.2x + 105 - 0.7x = 60
-0.5x = -45
x = 90
Therefore, the number of pints of the first type of fruit drink needed is 90, and the number of pints of the second type of fruit drink needed is 150 - 90 = 60.
Suppose that the sea level of an inlet is regularly measured at the same point on a bridge and
that high and low tides occur in equally spaced intervals. The high tide is observed to be 5 feet
above the average sea level of 10 feet; after 6 hours pass, the low tide occurs at 5 feet below
the average sea level.
x
In this task, you will model this occurrence using a trigonometric function by using as a
measurement of time. The first high tide occurs at x = = 3.
A. Identify the independent and dependent variables, both with letter names (x and y) and
what they represent in this scenario.
Independent variable:
Dependent variable:
B. Determine these key features of the function that models the tide (show/explain how
you found your values for each):
a. Amplitude:
b. Period:
C. Frequency
D. Midline
E. Vertical shift
F. Phase shift
C. Create a trigonometric function that models the ocean tide.
Explain why you chose your function type. Show work for any values not already outlined above.
D. Graph the function you wrote in part C
E. E. What is the height of the tide at × = 63? Show/explain your work.
For given problem, (A.) Independent variable is x, Dependent variable is y. (B.) Amplitude=5 feet, Period= 12 hours, frequency= 1/12 cycles per hour, midline is y= 10 feet, Vertical shift= 0, phase shift = 3 hours and trigonometric function that models the ocean tide is
[tex]y= 5 sin(\pi x/6-\pi /2)+10[/tex]
E. At x=63, Height of tide= y=10
How to identify dependent and Independent variable?A.Independent variable: x, representing time in hours since the first high tide occurred at x = 3.
Dependent variable: y, representing the height of the tide above or below the average sea level in feet.
How to determine key features of a function?B.
a. Amplitude: The amplitude is the distance from the average sea level to the highest point of the tide, or from the average sea level to the lowest point of the tide. In this scenario, the amplitude is 5 feet.
b. Period: The period is the time it takes for one complete cycle of the tide, from high tide to high tide or from low tide to low tide. In this scenario, the period is 12 hours (6 hours from high tide to low tide, and another 6 hours from low tide to high tide.
c. Frequency: The frequency is the number of cycles per unit of time. In this scenario, the frequency is 1/12 cycles per hour.
d. Midline: The midline is the horizontal line representing the average sea level. In this scenario, the midline is y = 10 feet.
e. Vertical Shift: The vertical shift is the amount that the graph of the function is shifted up or down from the midline. In this scenario, the vertical shift is 0, since the high tide is 5 feet above the average sea level and the low tide is 5 feet below the average sea level, so the average of the high and low tides is at the midline.
f. Phase Shift: The phase shift is the horizontal shift of the graph of the function. In this scenario, the first high tide occurs at x = 3, which is 3 hours after the starting point (x = 0). So the phase shift is 3 hours.
How to find trigonometric function modeling the ocean tide?C. Using the standard form of a trigonometric function, we can model the tide as:
[tex]y = A\; sin(2\pi ft -\phi ) + B[/tex]
where A is the amplitude, f is the frequency,[tex]\phi[/tex] is the phase shift, B is the vertical shift and the function 'sin' represents the sine of an angle in radians.
Substituting the values we found in part B, we get:
[tex]y = 5\;sin(2\pi \;(1/12)\;x-\pi/2) + 10[/tex]
Therefore, the trigonometric function that models the ocean tide is:
[tex]y= 5 sin(\pi x/6-\pi /2)+10[/tex]
D. Graph of [tex]y=5\sin(\pi\left(x/6\right)-\left(\pi/2)\right)+10[/tex] ( refer to image attached )
E. Height of tide at x=63,
[tex]y=5\sin(\pi\left(63/6\right)-\left(\pi/2)\right)+10[/tex]
[tex]y=5\sin(21\left(\pi/2\right)-\left(\pi/2)\right)+10\\\\y=5\sin(20\left(\pi/2\right))+10\\\\y=5\sin(10\left(\pi\right))+10\\\\y=10[/tex](∵Sin (nπ) = 0)
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Which choice is equivalent to the quotient shown here when x > 0?
√22x6 = √11x4
Answer:
Option B
Step-by-step explanation:
sqrt(22x^6) / sqrt(11x^4) = sqrt(2x^2)
sqrt(2x^2) can be rewritten as x sqrt(2)
Please answer in detail
Answer:
The area of triangle APB is 240 m².
Step-by-step explanation:
We are told that PA and PB are equal in length. Therefore, triangle APB is an isosceles triangle with base AB.
As PQ is perpendicular to the base (signified by the right angle symbol), point Q is the midpoint of AB. This means that:
The length of AQ is equal to the length of QB. PQ is the altitude (height) of triangle APB.As triangle PQB is a right triangle, use Pythagoras Theorem to find the length of QB:
[tex]\implies PQ^2+QB^2=PB^2[/tex]
[tex]\implies 24^2+QB^2=26^2[/tex]
[tex]\implies 576+QB^2=676[/tex]
[tex]\implies 576+QB^2-576=676-576[/tex]
[tex]\implies QB^2=100[/tex]
[tex]\implies \sqrt{QB^2}=\sqrt{100}[/tex]
[tex]\implies QB=10[/tex]
As AQ = QB, and QB is 10 m, then AQ is also 10 m.
Therefore, we can calculate the length of the base AB:
[tex]\begin{aligned}\implies AB &= AQ + QB\\&=10 + 10 \\&= 20\; \sf m\end{aligned}[/tex]
Now we have the length of the base of the triangle and the height of the triangle, we can calculate the area of triangle APB:
[tex]\begin{aligned}\textsf{Area of triangle $APB$}&=\dfrac{1}{2} \cdot \sf base \cdot height\\\\&=\dfrac{1}{2} \cdot AB \cdot PQ\\\\&=\dfrac{1}{2} \cdot 20 \cdot 24\\\\&=10 \cdot 24\\\\&=240\; \sf m^2\end{aligned}[/tex]
Therefore, the area of triangle APB is 240 m².
A small manufacturing company makes $125 on each stereo sound bar produces, and $100 profit on each flatscreen TV it makes. Each sound bar and TV must be processed by cutting machine (A), a fitting machine (B), and a polishing machine(C). Each sound bar must be processed on machine A for one hour, machine B for one hour, and machine C for four hours. Each TV must be processed or machine A for two hours, Machine B for one hour, and machine C for one hour. Machine A is available for 16 hours machine B 49 machine C for 24 hours.
An employee started a new job and must enroll in a new family health insurance plan. One of the options involves prescription drug coverage. The employee estimates that the entire family will fill 6 prescriptions per month, totaling $850. The monthly premium for the plan is $48, with 90% coverage for the first $450 in prescription costs, then 80% coverage for all prescription costs over $450. What is the total out-of-pocket expense for one month?
$173
$125
$773
$677
The total out-of-pocket expense for one month is $677. The correct option is D.
The first step is to calculate the amount of the prescription costs that will be covered by the insurance plan.
Since the family will fill 6 prescriptions per month and the total cost is $850, the average cost per prescription is $850/6 = $141.67 per prescription.
The insurance plan covers 90% of the first $450 in prescription costs, which is $450 * 0.9 = $405.
The remaining $141.67 - $405 = -$263.33 of the first prescription is not covered, since it is below the $450 threshold. This means that the family will have to pay the full cost of the first prescription, and the insurance will not contribute anything to it.
For the remaining 5 prescriptions, the insurance will cover 80% of the cost, since they are over the $450 threshold. The remaining 20% will be the family's responsibility.
The cost of the remaining 5 prescriptions is $141.67 x 5 = $708.35.
The insurance will cover 80% of this amount, which is $708.35 x 0.8 = $566.68.
The family will be responsible for the remaining 20% of the cost, which is $708.35 x 0.2 = $141.67.
Adding up all the costs, the total out-of-pocket expense for one month is:
$405 (first prescription) + $141.67 (20% of remaining prescription costs) + $48 (monthly premium) = $594.67
Therefore, the correct answer is option D: $677.
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In a class of 25 students, 7 students each have 2 brothers or sisters. 4 students each have 1 brother or sister. 3 students each have 0 brothers or sisters. 6 students each have 3 brothers or sisters. The rest of the students each have 4 brothers or sisters. The students make a line plot showing the numbers of brothers and sisters. Part A Use the drop-down menus to make the statement true. The scale of the line plot should have Choose... as its least data value and Choose... as its greatest data value. Part B Use the drop-down menus to make each of the statements about the number of dots true. There will be Choose... dot(s) above 1. There will be Choose... dot(s) above 4. There will be 3 dots above Choose... . 3 / 6 2 of 6 Answered
Step-by-step explanation:
Part A
There are 25 students in the class, and the least number of brothers or sisters is 0, so the least data value on the line plot should be 0. The greatest number of brothers or sisters is 4, so the greatest data value on the line plot should be 4.
Part B
There are 7 students with 2 brothers or sisters, so there will be 7 dots above 2. There are 4 students with 1 brother or sister, so there will be 4 dots above 1. There are 6 students with 3 brothers or sisters, so there will be 6 dots above 3. There are 3 students with 4 brothers or sisters, so there will be 3 dots above 4.
Therefore, the correct answers are:
Least data value: 0
Greatest data value: 4
Dots above 1: 4
Dots above 4: 7
Dots above 3: 6
please answer this question
The measure of the angle m∠RTS in the circle is drives to be equal to 105°.
What are angles on same segment of a circleAny two angles that have the same endpoint that lie on the same segment of the circle, that is the region between the chord joining the endpoints of the angle and the arc not containing the angle are said to be equal in measure.
Thus;
m∠QPT = m∠RST
so; m∠RST = 34°
Considering the triangle RST, we can solve for m∠RTS as follows:
m∠RTS = 180° - (34 + 41)° {sum of interior angles of a triangle}
m∠RTS = 180° - 75°
m∠RTS = 105°
Therefore, the measure of the angle m∠RTS in the circle is drives to be equal to 105°
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What is -3 5/7 x -2 1/2 Pleas help
me asap!!!
To multiply mixed numbers, we first need to convert them to improper fractions, then we can multiply them and simplify the result back to a mixed number.
Converting the first mixed number to an improper fraction:
-3 5/7 = -3 × 7/7 + 5/7 = -21/7 + 5/7 = -16/7
Converting the second mixed number to an improper fraction:
-2 1/2 = -2 × 2/2 + 1/2 = -4/2 + 1/2 = -3/2
Multiplying the two improper fractions:
(-16/7) × (-3/2) = (16/7) × (3/2) = 48/14 = 24/7
Simplifying the result to a mixed number:
24/7 = 3 3/7
Therefore, -3 5/7 x -2 1/2 = 3 3/7.
Question
3
Z
Note: Figure is not drawn to scale.
If y = 13 inches, z = 18 inches, h = 6 inches, and w = 4 inches, what is the area of the object?
OA.
180 square inches
OB.
120 square inches
OC. 252 square inches
OD. 126 square inches
The area of the figure is determined as 126 in².
option D.
What is the area of the figure?
The area of a shape describes the measure of the amount of space enclosed by a two-dimensional shape or surface. It is typically expressed in square units.
The area of the figure is calculated as follows;
Total area = area of triangle + area of rectangle
Area of triangle = ¹/₂ x base x height
Area of triangle = ¹/₂ x 18 in x 6 in
Area of triangle = 54 in²
Area of rectangle = 18 in x 4 in
Area of rectangle = 72 in²
Total area of the figure = 54 in² + 72 in² = 126 in²
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help use a net to find the surface area
The total surface area of the figures are 897.5in², 230cm², 217in² and 192in²
What is the net surface area of the figures?1. The net surface area of to the figure can be calculated using the formula;
TSA = (s + s + s)l * bh
s = side length
TSA = (15 + 17 + 8) * 17 + 14.5 * 15
h = 14.5 from Pythagorean theorem
Substituting the values into the formula
TSA = 897.5 in²
2. The total surface area of the figure is calculated as
TSA = (lw + wh + lh)
TSA = [(13 * 8) + (8 * 6) + (13 * 6)]
TSA = 230 cm²
3. The total surface area of the figure is
TSA = (lw + wh + lh)
TSA = [(11 * 2) + (2 * 15) + (15 * 11)]
TSA = 217 in²
4. The total surface area of the figure is
TSA = (lw + wh + lh)
TSA = (8 * 8) + (8 * 8) + (8 * 8)
TSA = 192in²
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A cylinder has a diameter of 8 cm and a height of 2 cm. What is the surface area of the cylinder?
The surface area of the cylinder is 150.7 cm^2.
The formula for calculating the surface area of the cylinder is 2πrh.
Here, r is the radius of the cylinder and h is the height of the cylinder.
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John has 4 liters of water on his counter. How many milliliters of water does John have on his counter?
Answer: 4000 millilitres
Step-by-step explanation:
One liter is equal to 1000 millilitres.
So since to get 4 from one, you multiply 1 x 4.
To get to the number of millilitres, you multiply 1000 x 4.
1000 x 4 is 4000.
Maricopa's Success scholarship fund receives a gift of $ 160000. The money is invested in stocks,
bonds, and CDs. CDs pay 5 % interest, bonds pay 4.1 % interest, and stocks pay 6.7 % interest.
Maricopa Success invests $ 30000 more in bonds than in CDs. If the annual income from the
investments is $ 8220, how much was invested in each account?
Maricopa Success invested $
Maricopa Success invested $
Maricopa Success invested $
in stocks.
in bonds.
in CDs.
Thus, Maricopa invested $60000 in stocks, $70000 in bonds, $40000 in CDs.
How to solve system of linear equations?A system of linear equations that can be solved using algebraic methods. In the provided problem, we have three unknowns (the amounts invested in CDs, bonds, and stocks) and three equations based on the supplied data (the total investment amount, the interest rates of each investment, and the total annual income from the investments).
We can ascertain the values of the unknowns, i.e. the amounts invested in each account, by setting up and solving these equations.
Let the principal amount invested in CDs be 'p' :
CDs:
Principal = p .......................(1)Rate = 5/100 = 0.05Interest = 0.05pBonds:
Principal = [tex]p + 30000[/tex] (given) ........................(2)Rate = 4.1/100 = 0.041Interest = [tex]0.041(p + 30000) = 0.041p + 1230[/tex]Stock:
Principal = [tex]160000 - p - (p + 30000) = 130000 - 2p[/tex] ............(3)Rate = 6.7/100 = 0.067Interest = [tex]0.067(130000 - 2p) = 8710 - 0.134p[/tex]Total interest:
[tex](0.05p) + (0.041p + 1230) + (8710 - 0.134p) = \$8220[/tex] (given)
[tex]-0.043p + 9940 =8220\\-0.043p = -1720[/tex]
Solving further for p:
p = $40,000
Putting value of p in (1), (2) and (3), we get;
[tex]CDs:\\Principal = p = \$40,000\\Bonds:\\Principal = p + 30000 = 40,000 + 30000 = \$70,000\\Stocks:\\Principal = 140000 - 2p = 140000 - 80000= \$60000\\[/tex]
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Which figure has at least one line of symmetry?
Responses
rectangle
Image with alt text: rectangle
right trapezoid
Image with alt text: right trapezoid
parallelogram
Image with alt text: parallelogram
I don't know.
a. The figure that has at least one line of symmetry is rectangle.
b. Another name for <XYZ in the figure is <Y.
What is a line of symmetry?A given straight line drawn from one edge to another edge of a figure in such a way that it divides a given figure into two equal parts with the same characteristics is known as the line of symmetry. Thus a given figure may have more than one line of symmetry, depending on the figure.
Considering the given figures in the question, we have;
ai. Rectangle: This is a quadrilateral in which opposite sides are equal and parallel. It has more than one line of symmetry.
ii. Right trapezoid: This is a trapezium in which one of its internal angles is a right angle. It has only one line of symmetry.
iii. Parallelogram: This is a quadrilateral in which the opposites are equal but slant, while the top and base are equal and not slant. It does not have a line of symmetry.
b. In the given figure, another name that can be given to <XYZ is <Y.
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"∠Y" is the answer of which you seek.
I want you guys to be able to answer your questions with assurance that my answer will work for you guys. I hope this helps you guys get 100% on that k12 diagnostic!
SCREENSHOT
Find the coordinate of point m
that partitions the segment Ab
In the ratio 3:4 if A 7,-3 and b -7,4
The coordinates of point M divides the line segment AB int he ratio 3 : 4 is given by ( 1, 0).
Ratio in which line segment AB divides m : n = 3 : 4
Let us consider the coordinates of A be ( x₁ , y₁ ) = ( 7, -3 )
Let us consider the coordinates of B be ( x₂ , y₂ ) = ( - 7 , 4)
Let us consider the coordinates of M be ( x , y )
Using the formula of line segment divides in the ration m: n we have,
x = ( mx₂ + nx₁ ) / ( m + n )
y = ( my₂ + ny₁ ) / ( m + n )
Substitute the values we have,
x = ( 3 × (-7) + 4 × 7 ) / ( 3 + 4 )
= ( -21 + 28 ) / 7
= 1
y = ( 3 × (4) + 4 × -3 ) / ( 3 + 4 )
= ( 12 - 12 ) / 7
= 0
Therefore, the coordinates of point M which partition the line segment AB is equal to ( 1, 0).
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The above question is incomplete, the complete question is:
Find the coordinate of point M that partitions the segment AB in the ratio 3:4,if A (7,-3) and B (-7,4).
What is the sample space Aryonna rolls a standard number cube once
S = {1, 2, 3, 4, 5, 6} is the sample space Aryonna rolls a standard number cube once
The sample space for rolling a standard number cube once consists of all the possible outcomes or numbers that could appear on the face of the cube.
Since a standard number cube has six faces, numbered 1 through 6, the sample space is:
S = {1, 2, 3, 4, 5, 6}
Each element in this set represents a possible outcome of rolling the number cube once.
Therefore, S = {1, 2, 3, 4, 5, 6} is the sample space Aryonna rolls a standard number cube once
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A circular plate has a radius of 7 inches. Which is closest to the area of the plate?
Answer:
Approximately [tex]153.94[/tex] [tex]in^2[/tex]
Step-by-step explanation:
Recall the formula for the area of a circle:
[tex]\pi r^2[/tex]
Where r is the radius.
We are given that the radius is 7 inches. Let's plug that into our formula and solve it. We have:
[tex]\pi r^2=\\\pi (7)^2=\\\pi (49)=\\49\pi[/tex]
From here, we multiply our answer by pi using a calculator and round.
Using a calculator, we can find that [tex]49\pi[/tex] is approximately 153.93804 (round to the nearest hundredth: 153.94)Simplify the expression using the properties of exponents. Expand any numerical portion of your answer and only include positive exponents.
(5m^3n^3)^2/n
The simplified form of expression [tex]\frac{(5m^3n^3)^2}{n}[/tex] is [tex]25m^6n^5[/tex]
Consider the expression [tex]\frac{(5m^3n^3)^2}{n}[/tex]
We know that the exponent rule that [tex](a\times b)^m=a^m \times b^m[/tex]
and [tex](a^m)^n=a^{m\times n}[/tex]
where a, b, m and n real numbers.
Consider the numerator of above expression.
[tex](5m^3n^3)^2[/tex]
Using the rule [tex](a\times b)^m=a^m \times b^m[/tex] ,
[tex](5\times m^3\times n^3)^2\\\\= 5^2\times (m^3)^2\times (n^3)^2\\[/tex]
We know that the square of 5 is 25
so, 5² = 25
Now we use the rule [tex](a^m)^n=a^{m\times n}[/tex]
so, above expression becomes,
[tex]5^2\times (m^3)^2\times (n^3)^2\\\\=25\times m^{3\times 2}\times n^{3\times 2}\\\\=25\times m^6 \times n^6[/tex]
Also, we know that inverse of any value can be written as [tex]\frac{1}{a} =a^{-1}[/tex]
So, using this rule of exponent the value of 1/n would be [tex]n^{-1}[/tex]
So, the expression [tex]\frac{(5m^3n^3)^2}{n}[/tex] becomes,
[tex]\frac{(5m^3n^3)^2}{n}\\\\=\frac{25\times m^6 \times n^6}{n}\\\\=(25\times m^6 \times n^6)\times n^{-1}\\\\=25\times m^6 \times n^6\times n^{-1}[/tex]
We know that if the base of exponents are same then we add the powers of exponents.
So, our expression becomes,
[tex]=25\times m^6\times n^{6-1}\\\\=25\times m^6\times n^5[/tex]
This is the required expression.
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Suppose that the functions f and g are defined as follows.
Find f.g and f+ g. Then, give their domains using interval notation.
The composite functions and their domains are (f . g)(x) = √(4x - 1)/(5x² + 3) and [tex](f + g)(x) = \frac{5x^2\sqrt{4x-1}+3\sqrt{4x-1}+1}{5x^2+3}[/tex]
Domain: x ≥ 1/4
Calculating the composite functions and their domainsGiven that
f(x) = 1/(5x² + 3)
g(x) = √(4x - 1)
For the first pair, we have
(f . g)(x) = f(x) * g(x)
So, we have
(f . g)(x) = 1/(5x² + 3) * √(4x - 1)
Evaluate
(f . g)(x) = √(4x - 1)/(5x² + 3)
For the domain, we have
4x - 1 ≥ 0
x ≥ 1/4
Next, we have
(f . g)(x) = f(x) * g(x)
So, we have
(f + g)(x) = 1/(5x² + 3) + √(4x - 1)
Evaluate
[tex](f + g)(x) = \frac{5x^2\sqrt{4x-1}+3\sqrt{4x-1}+1}{5x^2+3}[/tex]
For the domain, we have
4x - 1 ≥ 0
x ≥ 1/4
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Solve the following equation for the unknown quantity x.
[tex]\mathrm{9^x=40}[/tex]
*Round to the nearest hundredth as needed*
Answer:
[tex]x = 1.68[/tex]
Step-by-step explanation:
We have the equation
9ˣ = 40 and we have to solve for x
Take logs on both sides
[tex]\ln(9^x) = \ln(40)\\\\\ln(9^x) = x ln(9) \\\\\therefore x \ln(9) = \ln(40)\\\\x = \dfrac{\ln(40)}{\ln(9)}\\\\\text{Using a calculator, this works out to: }\\\\x=1.67888139\dots \\\\\text{Rounded to the nearest hundredth, }\\x = 1.68\\\\[/tex]
The brightness of illumination, l, of an object varies inversely as the square of its distance , d, from the source of illumination, if l = 18 luxes when d= 4m, find l if d =3m
The value of source of illumination (I) is 32 luxes when the distance d = 4 m.
The brightness of illumination that is denoted by I of ab object varies inversely as the square of of its distance which is denoted by variable 'd'.
So, I ∝ 1/d²
Therefore, I = k/d².......... (i), where k is a non zero proportionality constant.
It is given that if the source of illumination I = 18 luxes when d = 4 m.
Substituting I = 18 and d = 4 in the equation (i) we get,
18 = k/4²
k/16 = 18
k = 18*16 = 288
Now we have to find the value of I when d = 3m.
Now the relation becomes. I = 288/d²
Substituting the value d = 3 in equation (ii) we get,
I = 288/3² = 288/9 = 32 Iuxes.
Hence the source of illumination now is 32 Iuxes.
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