The formula you need to use in order to find the volume of a cylinder is [tex]V = \pi r^{2} h[/tex]
Step 1) Take note of the information that has been provided to you.
The height is 8 yards
The radius is 2 years
Step 2) Substitute the information above into the formula.
[tex]V = \pi (2^{2})(8)[/tex]
Step 3) Solve.
[tex]V = \pi (2^{2})(8)[/tex] [tex]=100.53[/tex]
Step 4) Round if necessary.
[tex]100.53 = 101[/tex]
Whish of the following are necessary when proving that the angles of a parallelogram are congruent
D. Angle Addition Postulate is necessary when proving that the angles of a parallelogram are congruent.
What is parallelogram?
A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measure.
To prove that opposite angles of a parallelogram are congruent, we need to use the properties of parallelograms, such as opposite sides are parallel and congruent, opposite angles are congruent, consecutive angles are supplementary, and diagonals bisect each other.
We do not need the Segment Addition Postulate, as it is used to find a length of a segment, not to prove the congruence of angles.
We also do not need to use the Opposite sides are perpendicular property, as this is only true for a rectangle or a rhombus, but not necessarily for a parallelogram.
Similarly, the Angle Addition Postulate is used to find the measure of an angle, not to prove that angles are congruent.
Therefore, the only necessary property to prove that opposite angles of a parallelogram are congruent is that opposite angles are congruent, which is a property of parallelograms.
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Speed limit of 45 mph is equivalent to 72 km an hour. a sign says speed limit is 90 km an hour. what is the speed limit in miles per hour?
The required answer is the speed limit of 90 km/h is equivalent to approximately 56 mph.
To convert the speed limit from kilometers per hour (km/h) to miles per hour (mph), use the conversion factor of 1 km/h = 0.621371 mph. Let's follow these steps:
Step 1: Determine the conversion factor.
Since we know the conversion factor is 1 km/h = 0.621371 mph, use this to convert the speed limit from km/h to mph.
Step 2: Calculate the speed limit in mph.
The given speed limit is 90 km/h. Multiply this value by the conversion factor to obtain the equivalent speed in mph:
90 km/h x 0.621371 mph/km/h = 55.92339 mph
Step 3: Round the result, if necessary.
To provide a practical speed limit, we can round the result to the nearest whole number. Therefore, the speed limit in miles per hour is approximately 56 mph.
Hence, the speed limit of 90 km/h is equivalent to approximately 56 mph.
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can you solve this question?
dy/dx=?
The solution of the equation is: dy/dx = -3[tex]e^{3x}[/tex]/ √(1-[tex]e^{6x}[/tex])
What is the solution?
Let's start by using the chain rule to find dy/dx:
dy/dx = dy/du * du/dx
where u = [tex]e^{3x}[/tex]
We can find du/dx using the power rule:
du/dx = 3[tex]e^{3x}[/tex]
Now we need to find dy/du. We can use the derivative of arccos function:
dy/du = -1/√(1-u²)
Substituting u = [tex]e^{3x}[/tex], we get:
dy/dx = dy/du * du/dx
dy/dx = -1/√(1-[tex]e^{6x}[/tex]) * 3[tex]e^{3x}[/tex]
So the final answer is:
dy/dx = -3[tex]e^{3x}[/tex] / √(1-[tex]e^{6x}[/tex])
what is chain rule?
The chain rule is a fundamental rule of calculus that allows you to find the derivative of a composite function. A composite function is a function that is formed by taking one function and plugging it into another function.
For example, if we have two functions f(x) and g(x), the composite function is defined as:
h(x) = f(g(x))
To find the derivative of the composite function h(x), we use the chain rule, which states that:
h'(x) = f'(g(x)) * g'(x)
In words, this means that to find the derivative of the composite function, we first take the derivative of the outer function f(x) with respect to its input, evaluated at the inner function g(x), and then multiply it by the derivative of the inner function g(x) with respect to its input.
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Complete question is: dy/dx = -3[tex]e^{3x}[/tex]/ √(1-[tex]e^{6x}[/tex])
I cant seem to figure out this answer. Can anyone help?
A company makes wax candles in the shape of a solid sphere. Suppose each candle has a diameter of 18 cm. If the company has a total of 152,604 cm³ of w
how many candles can be made?
Use 3.14 for x, and do not round your answer.
Answer:
50 spherical candles can be made with 152,604 cm³ of wax.
Step-by-step explanation:
Since the wax candles are in the shape of a solid sphere, we can calculate the volume of one candle by using the volume of a sphere formula.
[tex]\boxed{\begin{minipage}{4 cm}\underline{Volume of a sphere}\\\\$V=\dfrac{4}{3} \pi r^3$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]
The diameter of a sphere is twice its radius.
Therefore, if the diameter of the spherical candle is 18 cm, its radius is:
[tex]\implies r=\dfrac{d}{2}=\dfrac{18}{2}=9\; \sf cm[/tex]
Substitute r = 9 and π = 3.14 into the formula to calculate the volume of one spherical candle.
[tex]\begin{aligned}\implies \textsf{Volume of one candle}&=\sf \dfrac{4}{3} \cdot 3.14 \cdot (9\;cm)^3\\\\&=\sf \dfrac{4}{3} \cdot 3.14 \cdot 729\;cm^3\\\\&=\sf 3052.08\; \sf cm^3\end{aligned}[/tex]
Given the company has a total of 152,604 cm³ of wax, to calculate how many candles can be made, divide the total amount of available wax by the wax needed to make one candle.
[tex]\begin{aligned}\textsf{Total number of candles}&=\sf \dfrac{152604\; cm^3}{3052.08 \;cm^3}\\\\&=\sf 50\end{aligned}[/tex]
Therefore, 50 spherical candles can be made with 152,604 cm³ of wax.
can someone help me please
A car was valued at $44,000 in the year 1992. The value depreciated to $15,000 by the year 2006.
A) What was the annual rate of change between 1992 and 2006?
r=---------------Round the rate of decrease to 4 decimal places.
B) What is the correct answer to part A written in percentage form?
r=---------------%
C) Assume that the car value continues to drop by the same percentage. What will the value be in the year 2009 ?
value = $ -----------------Round to the nearest 50 dollars.
(A) the annual rate of change between 1992 and 2006 was 0.0804
(B) r = 0.0804 * 100% = 8.04%
(C) value in 2009 = $11,650
What is the rate of change?
The rate of change is a mathematical concept that measures how much one quantity changes with respect to a change in another quantity. It is the ratio of the change in the output value of a function to the change in the input value of the function. It describes how fast or slow a variable is changing over time or distance.
A) The initial value is $44,000 and the final value is $15,000. The time elapsed is 2006 - 1992 = 14 years.
Using the formula for an annual rate of change (r):
final value = initial value * [tex](1 - r)^t[/tex]
where t is the number of years and r is the annual rate of change expressed as a decimal.
Substituting the given values, we get:
$15,000 = $44,000 * (1 - r)¹⁴
Solving for r, we get:
r = 0.0804
So, the annual rate of change between 1992 and 2006 was 0.0804 or approximately 0.0804.
B) To express the rate of change in percentage form, we need to multiply by 100 and add a percent sign:
r = 0.0804 * 100% = 8.04%
C) Assuming the car value continues to drop by the same percentage, we can use the same formula as before to find the value in the year 2009. The time elapsed from 2006 to 2009 is 3 years.
Substituting the known values, we get:
value in 2009 = $15,000 * (1 - 0.0804)³
value in 2009 = $11,628.40
Rounding to the nearest $50, we get:
value in 2009 = $11,650
Hence, (A) the annual rate of change between 1992 and 2006 was 0.0804
(B) r = 0.0804 * 100% = 8.04%
(C) value in 2009 = $11,650
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A pilot is flying over a straight highway. He determines the angles of depression to two mileposts,
6.7 km apart, to be 38° and 41°, as shown in the figure.
A
NOTE: The picture is NOT drawn to scale.
38°
6.7 km
41°
B
What is the elevation of the plane in meters? Give your answer to the nearest whole number.
height=
meters
Answer:
a) 4.46 miles
b) 3 miles
Step-by-step explanation:
Law of Sines:
[tex]\dfrac{\text{a}}{\text{sin(A)}} =\dfrac{\text{b}}{\text{sin(B)}}[/tex]
a) The distance of the plane from point A
The angle of depression corresponds to the congruent angle of elevation therefore, 180 - 28 - 52 = 100°
[tex]\dfrac{\text{6.7}}{\text{sin(100)}} =\dfrac{\text{b}}{\text{sin(41)}}[/tex]
[tex]\text{b}=\dfrac{6.7\text{sin}(41)}{\text{sin}(100)}[/tex]
[tex]\text{b}=4.46 \ \text{miles}[/tex]
b) Elevation of the plane
[tex]\text{sin}=\dfrac{\text{opposite}}{\text{hypotenuse}}[/tex]
hypotenuse is 4.46 and opposite is the elevation(h) to be found
[tex]\text{sin}(38)=\dfrac{\text{h}}{4.46}[/tex]
[tex]\text{h}=\text{sin}(38)4.46[/tex]
[tex]\text{h}=3[/tex]
What is the Surface Area of the Triangular Prism below?
Surface area is calculated as 48 + 120 = 168 square units (area of triangular faces + area of rectangular faces).
A triangular prism is what?A polyhedron with two triangular sides and three rectangles sides is referred to as a triangular prism. It is a three-dimensional shape with two base faces, three side faces, and connections between them at the edges.
Given :
We must calculate the area of each face of the triangular prism and put them together to determine its surface area.
The areas of the triangular faces are equal, so we may calculate one of their areas and multiply it by two:
One triangular face's area is equal to (1/2) the sum of its base and height, or (1/2) 6 x 8 x 6, or 24 square units.
Both triangular faces' surface area is 2 x 24 or 48 square units.
Finding the area of the rectangular faces is now necessary:
One rectangular face's area is given by length x breadth (10 x 6) = 60 square units.
120 square units are the area of both rectangular faces or 2 by 60.
Hence, the triangular prism's total surface area is:
Surface area is calculated as 48 + 120 = 168 square units (area of triangular faces + area of rectangular faces).
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how create the own congruent postulate sss
Step-by-step explanation:
The SSS (Side-Side-Side) Congruence Postulate states that if three sides of a triangle are congruent to three sides of another triangle, then the triangles are congruent.
To create your own SSS Congruence Postulate, you could use the following statement:
"If the corresponding sides of two triangles are congruent, then the triangles are congruent."
This postulate would encompass the SSS Postulate, as well as two other postulates: the SAS (Side-Angle-Side) Postulate and the ASA (Angle-Side-Angle) Postulate.
Using this postulate, you could show that two triangles are congruent if, for example, their corresponding sides are all 5cm long. This would mean that the triangles have the same shape and size, even if they are oriented differently.
Hopes this helps
need help with this problem
Answer:
area=28cm^2
Step-by-step explanation:
4×5=20
20÷2=10 area of the bigger triangle
4×2=8
8÷2=4 area of the smaller triangle
10+10=20 for area of the the 2 bigger triangles
4+4=8 for the area of the 2 smaller triangles
area of kite= 20+8= 28
PLEASE HELP SOLVE ALL !!!!!
Step-by-step explanation:
6.
the figure can be considered as a combination of a 16×8 rectangle and a 10×8 right-angled triangle on the left side.
for both, perimeter and area, we need to find the length of the missing third side of the right-angled triangle, as this is a part of the long baseline of the overall figure.
as it is a right-angled triangle, we can use Pythagoras :
c² = a² + b²
"c" being the Hypotenuse (the side opposite of the 90° angle, in our case 10 ft). "a" and "b" being the legs (in our case 8 ft and unknown).
10² = 8² + (leg2)²
100 = 64 + (leg2)²
36 = (leg2)²
leg2 = 6 ft
that means the bottom baseline is 16 + 6 = 22 ft long.
a.
Perimeter = 10 + 16 + 8 + 22 = 56 ft
b.
Area is the sum of the area of the rectangle and the area of the triangle.
area rectangle = 16×8 = 128 ft²
area triangle (in a right-angled triangle the legs can be considered baseline and height) = 8×6/2 = 24 ft²
total Area = 128 + 24 = 152 ft²
7.
it was important that the bottom left angle of the quadrilateral was indicated as right angle (90°). otherwise this would not be solvable.
but so we know, it is actually a rectangle.
that means all corner angles are 90°.
therefore, the angle AMT = 90 - 20 = 70°.
the diagonals split these 90° angles into 2 parts that are equal in both corners of the diagonal, they are just up-down mirrored.
a.
so, the angle HTM = AMT = 70°.
b.
MEA is an isoceles triangle.
so, the angles AME and EAM are equal.
the angle AME = AMT = EAM = 70°.
the sum of all angles in a triangle is always 180°.
therefore,
the angle MEA = 180 - 70 - 70 = 40°
c.
both diagonals are equally long, and they intersect each other at their corresponding midpoints.
so, when AE = 15 cm, then AH = 2×15 = 30 cm.
TM = AH = 30 cm.
3(26+14)/(2x2) i need help please
Answer:
30
Step-by-step explanation:
its right
order of operations
Answer:
The answer is 30...
Step-by-step explanation:
Apply the rule BODMAS...
3(40)/(4)
120/4
30
identify the two forms of the simplified expression: x^5/x^10
Use the words exponent, numerator (top) and denominator (bottom) in your description
Please help.
The expression [tex]x^{5}[/tex]/[tex]x^{10}[/tex] can be written in two forms: Exponent form, which is simplified and has a base of x, and Fraction form, which has a numerator and denominator separated by a horizontal line.
What is an exponent form?Exponents are a shortened notation for repeated multiplication of a number, and exponent form is a means of formulating a mathematical equation using exponents. A superscript (a little raised number) is written to the right of a number or variable that has been raised to a power in exponent form.
There are two distinct but equal ways to write the expression "[tex]x^{5}[/tex]/[tex]x^{10}[/tex]":
Exponent form: In this format, the exponents of the equation are consolidated and made simpler. We may subtract the exponents in the denominator from the exponents in the numerator since the bases of the numerator and denominator are both x:
[tex]x^5/x^10 = x^(5-10) = x^(-5)[/tex]
Fraction form: In this format, the equation is expressed as a fraction with a horizontal line between the numerator and denominator. X is increased to the fifth power in the denominator and to the tenth power in the numerator:
[tex]x^5/x^10[/tex]
X is increased to the fifth power, or x, in the numerator. X is multiplied by 10 to create the denominator, or [tex]x^{10}[/tex].
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CAN someone please help me with this question please?!!!! Worth 30 points
The total surface area of the given hemispherical scoop is: 30.41 cm²
How to find the surface area?The formula for the total surface area of a hemisphere is:
TSA = 3πr² square units
Where:
π is a constant whose value is equal to 3.14 approximately.
r is the radius of the hemisphere.
Since the steel is 0.2cm thick and the outside of the scoop has a radius of 2cm, then we can say that:
TSA = 3π(2 + 0.2)²
= 3π(2.2)²
= 30.41 cm²
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What A can do in 3 days, B can do in 4 days. If C takes 6 days to do a job that B can do in 5 days, how many days will it take A to do a job that C can do in 16 days?
I don’t know how to find the side length and the word problems are confusing(don’t worry about the ones I did).
The side of the given square is 36W^2 + 12W +1 of is 6W +1 and 81W^2 -72W + 16 is (9W-4).
How to calculate the area of the square?The area is calculated by multiplying the length of a shape by its width.
and the unit of the square is a square unit.
Given Area of the square :
1) [tex]36W^{2}+12W+1[/tex]
Area of square = [tex]36W^{2}+12W+1[/tex]
[tex]side^{2}[/tex] = [tex]36W^{2} + 6W +6W +1\\6W (6W +1) +1 (6W +1)\\(6W +1)(6W+1)\\(6W +1 )^{2} \\side^{2} = (6W + 1)^{2} \\side = 6W + 1[/tex]
[tex]Area of the square = 81W^{2} - 72W + 16\\(side)^{2} = (9W-4)(9W-4)\\(side)^{2} = (9W-4)^{2} \\side = (9W-4)[/tex]
Therefore the side of the square is 6W+1 and 9x-4.
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one fourth of a number is no less than -3
The solution to the problem is that the number "x" must be greater than or equal to -12.
What solution is provided for "1/4 of number not less than -3"?Let's define the variable as "x," representing the unknown number. According to the problem statement, one fourth of the number is no less than -3.
Mathematically, we can express this statement as:
x/4 ≥ -3
To solve for "x," we can start by multiplying both sides of the inequality by 4 to eliminate the fraction:
x ≥ -3*4
x ≥ -12
Therefore, the solution to the problem is that the number "x" must be greater than or equal to -12.
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The function in the table is quadratic:
x f(x)
-1 1/3
0 1
1 3
2 9
The quadratic function that fits the given points is: f(x) = (4/3)x² - (10/3)x + 1
By using the simplification formula
f(x) = ax² + bx + c
where a, b, and c are constants
a(-1)² + b(-1) + c = 1/3
a(0)² + b(0) + c = 1
a(1)² + b(1) + c = 3
a(2)² + b(2) + c = 9
Simplifying each
a - b + c = 1/3
c = 1
a + b + c = 3
4a + 2b + c = 9
We can solve this using any method like substitution, elimination
a - b + c = 1/3
a + b + c = 3
2a + 2c = 9/3
Adding the first two equations 2a + 2c = 10/3
Subtracting the third equation
b = 5a/3 - 2
a - (5a/3 - 2) + 1 = 1/3
a = 4/3
Finally, we can substitute a = 4/3 and b = 5a/3 - 2, and c = 1 into the standard form of the quadratic equation to get: f(x) = (4/3)x² - (10/3)x + 1.
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A block of wood has the shape of a triangular prism. The bases are right triangles. Find its surface area
The formula used to calculate the surface area of a triangular prism is:
S = bh + 2ah
Where S = surface area, b = length or side of the triangle, and h = height of the triangle.
For right triangles, the height can be calculated as:
c = a^(2) \+ b^(2)
c = the hypotenuse
a and b = the two sides of the triangle
Margo borrows $1400, agreeing to pay it back with 5% annual interest after 17 months. How much interest will she pay? Round your answer to the nearest cent, if necessary.
well, let's keep in mind that since a year has 12 months, then 17 months is really 17/12 of a year, so
[tex]~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\dotfill & \$1400\\ r=rate\to 5\%\to \frac{5}{100}\dotfill &0.05\\ t=years\dotfill &\frac{17}{12} \end{cases} \\\\\\ I = (1400)(0.05)(\frac{17}{12}) \implies I \approx 99.17[/tex]
Determine the simple interest. The rate is an annual rate. Assume 360 days in a year. p=$586.21, r=6.3%, t=83 days
so we're assuming there are 360 days in a year, so 83 days is really just 83/360 of a year, so
[tex]~~~~~~ \textit{Simple Interest Earned} \\\\ I = Prt\qquad \begin{cases} I=\textit{interest earned}\\ P=\textit{original amount deposited}\dotfill & \$586.21\\ r=rate\to 6.3\%\to \frac{6.3}{100}\dotfill &0.063\\ t=years\dotfill &\frac{83}{360} \end{cases} \\\\\\ I = (586.21)(0.063)(\frac{83}{360}) \implies I \approx 8.51[/tex]
The function g(x) is the height of a football x seconds after it is thrown in the air. The football reaches its maximum height of 28 feet in 6 seconds, and hits the ground at 12 seconds.
What is the practical domain for the function f(x)?
Type your answer in interval notation.
It can be expressed in interval notation as:[0, 12]
The practical domain for the function g(x) is [0,12], as this is the time during which the football is in the air, from the time it is thrown until it hits the ground.
The practical domain for the function g(x) would be the time interval during which the football is in the air, since it only makes sense to talk about the height of the football while it is airborne.
From the problem statement, we know that the football is thrown in the air at time x = 0, reaches its maximum height of 28 feet at time x = 6, and hits the ground at time x = 12. Therefore, the practical domain for the function g(x) is:
0 <= x <= 12
This means that the function g(x) is defined and meaningful for any value of x between 0 and 12, inclusive. Beyond this domain, the function does not have a practical interpretation because the football is either not yet thrown or has already hit the ground.
The function g(x) is the height of a football x seconds after it is thrown in the air. The football reaches its maximum height of 30 feet in 4 seconds, and hits the ground at 10 seconds.
What is the practical domain for the function f(x)
Write your answer in interval notation.
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T
√
Guided Instruction: Measures of Arcs and Central Angles
X
A
Undo (Ctrl + Z)
Circle G has radii GH and GK. Find the value of x if
m2KGH = 122° and mKH = (5x + 7)⁰.
3
Submit
15 of 21
On solving the provided question we can say that As a result, the value angles of x is 23.
what are angles?An angle is a shape in Euclidean geometry that is made up of two rays that meet at a point in the middle known as the angle's vertex. Two rays may combine to form an angle in the plane where they are located. When two planes collide, an angle is generated. They are referred to as dihedral angles. An angle in plane geometry is a possible configuration of two radiations or lines that express a termination. The word "angle" comes from the Latin word "angulus," which meaning "horn." The vertex is the place where the two rays, also known as the angle's sides, meet.
m(arc GH) = 122° plus 5x plus 7
5x + 129 = m(arc GH)
When we plug this into the equation we discovered earlier, we get:
5x + 7 = ½
(5x + 129)
10x + 14 = 5x + 129
5x = 115
x = 23
As a result, the value of x is 23.
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Which steps can be used to solve for the value of y?
2/3 (y+57)=178
A. Divide both sides by 2/3, then subtract 57 from both sides.
B. Subtract 57 from both sides, then divide both sides by 2/3.
C. Multiply both sides by 2/3, then subtract 57 from both sides.
D. Subtract 2/3 from both sides, then subtract 57 from both sides.
3
Select the correct answer.
Which phase of the business cycle would be marked by an increase in productivity while employment and profits also rise?
Answer:
The phase of the business cycle that would be marked by an increase in productivity while employment and profits also rise is known as the Expansion phase.
HELP ASP!!!
Ramon is filling cups with juice. Each cup is shaped like a cylinder and has a diameter of 4.2 inches and a height of 7 inches. How much juice can Ramon pour into 6 cups? Round to the nearest hundredth and approximate using π = 3.14.
96.93 cubic inches
553.90 cubic inches
581.59 cubic inches
2,326.36 cubic inches
Answer:
581,58 in^3
Step-by-step explanation:
Given:
Cylinder shaped cups
d (diameter) = 4,2 in
r (radius) = 0,5 × 4,2 = 2,1 in
h (height) = 7 in
π = 3.14
.
First, let's find how much juice can he pour into 1 cup:
.
We need to find the base of the cylinder:
.
[tex]a(base) = \pi {r}^{2} = 3.14 \times( {2.1})^{2} = 13.8474[/tex]
.
Now, we can find the volume of one cup:
V = a (base) × h
[tex]v = 13.8474 \times 7 ≈96.93[/tex]
Multiply this number by 6 and we'll get the answer (since there's 6 cups):
96,93 × 6 = 581,58
A volcano on a recently discovered planet rises to a height of 69,657.652 ft.
Use the table of facts to find the height of the volcano in miles.
Round your answer to the nearest tenth.
Answer:
13.2 miles
Step-by-step explanation:
We can use the following conversion factors:
1 mile = 5,280 feet
Using this conversion, we can divide the height of the volcano in feet by 5,280 to get the height in miles:
69,657.652 ft ÷ 5,280 ft/mi ≈ 13.2 mi
Therefore, the height of the volcano on the recently discovered planet is approximately 13.2 miles.
Hopes this helps
Helen pulls one marble out of the box at random, records its color, replaces it, and mixes up the marbles again. If she does this 200 times, how many blue marbles should she excpect to pull out?
1/2 of the marbles are yellow
1/8 of the marbles are red
The rest of the marbles are blue
Answer:
The answer to your problem is, 75
Step-by-step explanation:
Yellow = [tex]\frac{1}{2}[/tex] of the marbles
Red = [tex]\frac{1}{8}[/tex] of the marbles
Blue = [tex]1 - \frac{1}{2} - \frac{1}{8} = \frac{1}{2} - \frac{1}{8}[/tex]
= [tex]\frac{1*4}{2*4} - \frac{1}{8}[/tex]
= [tex]\frac{1}{2} or \frac{4}{8} - \frac{1}{8}[/tex]
= [tex]\frac{3}{8}[/tex]
[tex]\frac{3}{8}[/tex] of the marbles are represented as blue.
200 x [tex]\frac{3}{8}[/tex] = 25 x 3 = 75
Thus the answer to your problem is, 75
Select the correct answer.
Which statement about the end behavior of the logarithmic function f(x) = log(x + 3) – 2 is true?
A.
As x decreases to the vertical asymptote at x = -3, y decreases to negative infinity.
B.
As x decreases to the vertical asymptote at x = -1, y decreases to negative infinity.
C.
As x decreases to the vertical asymptote at x = -3, y increases to positive infinity.
D.
As x decreases to the vertical asymptote at x = -1, y increases to positive infinity.
the correct answer is A: as x decreases to the vertical asymptote at x = -3, y decreases to negative infinity.
To determine the end behavior of the logarithmic function f(x) = log(x + 3) - 2, we need to look at what happens to the function as x approaches positive and negative infinity.
As x approaches negative infinity, the argument of the logarithm, (x + 3), becomes more and more negative. However, since logarithms are not defined for negative arguments, we need to shift the graph of the function to the left by 3 units to avoid the undefined region. This means that the vertical asymptote of the function is at x = -3. As x approaches -3 from the left, the argument of the logarithm becomes smaller and smaller negative numbers. However, the logarithm of a small negative number is a large negative number. Therefore, as x approaches -3 from the left, the function f(x) = log(x + 3) - 2 decreases to negative infinity. This eliminates options C and D.
As x approaches positive infinity, the argument of the logarithm, (x + 3), becomes more and more positive. Therefore, as x approaches positive infinity, the logarithm of (x + 3) becomes larger and larger. This means that the function f(x) = log(x + 3) - 2 approaches infinity as x approaches positive infinity. However, it approaches infinity from below since we subtract 2 from the logarithmic value.
To summarize, as x approaches -3 from the left, f(x) approaches negative infinity, and as x approaches positive infinity, f(x) approaches infinity from below. Therefore, the correct answer is A: as x decreases to the vertical asymptote at x = -3, y decreases to negative infinity.
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Write the following as an algebraic expression in u, u > 0. sin (artan u/ square root 3)
Answer: Let's start by using the identity:
tan(arctan(x)) = x
to simplify the expression inside the sine function. So, we have:
arctan(u) / sqrt(3) = tan(arctan(u) / sqrt(3))
Now, using the trigonometric identity:
tan(x/y) = sin(x) / (cos(y) + sin(y))
with x = arctan(u) and y = sqrt(3), we get:
tan(arctan(u) / sqrt(3)) = sin(arctan(u)) / (cos(sqrt(3)) + sin(sqrt(3)))
Simplifying further, we know that:
sin(arctan(u)) = u / sqrt(1 + u^2)
and
cos(sqrt(3)) + sin(sqrt(3)) = 2cos(sqrt(3) - pi/4)
So, the expression becomes:
sin(arctan(u) / sqrt(3)) = u / sqrt(1 + u^2) / [2cos(sqrt(3) - pi/4)]
Simplifying the denominator, we have:
sin(arctan(u) / sqrt(3)) = u / sqrt(1 + u^2) / (2(cos(sqrt(3))cos(pi/4) + sin(sqrt(3))sin(pi/4)))
Using the values for cosine and sine of pi/4, we get:
cos(pi/4) = sin(pi/4) = 1/sqrt(2)
So, we have:
sin(arctan(u) / sqrt(3)) = u / sqrt(1 + u^2) / [2(sqrt(3)/2 + 1/2)]
Simplifying further:
sin(arctan(u) / sqrt(3)) = u / (sqrt(1 + u^2) * (sqrt(3) + 1))
Therefore, the algebraic expression for sin(arctan(u) / sqrt(3)) is:
u / (sqrt(1 + u^2) * (sqrt(3) + 1))
Step-by-step explanation:
An expression is shown.
x³ + 2x² - 7x + 3x² + x³ - X
Given that x does not = 0, which of the following is equivalent to the expression?
Select one:
A. 2x³ + 5x² - 8x
B. −x^12
C. x³ + x²- x
D. x²+ x -1
Answer:
A: [tex]2x^3+5x^2-8x[/tex]
Step-by-step explanation:
In order to get the answer to this, you have to combine like terms to simplify the answer.
Go through and organize the x's by the size of the exponents. (this step isn't necessary but it can help you visualize it if you are having trouble with that)
[tex]x^3+x^3+2x^2+3x^2-7x-x[/tex]
When the variables are raised to the same degree, the coefficients can be added together.
[tex]2x^3+5x^2-8x[/tex]