Write the ratio of corresponding sides for the similar triangles and reduce the ratio to lowest terms
To find the ratio of corresponding sides for similar triangles, compare the lengths of the corresponding sides in both triangles. If the triangles are similar, these ratios will be equal. To simplify the ratio, divide each term by the greatest common factor.
Explanation:In mathematics, to write the ratio of corresponding sides for similar triangles, you need to compare the lengths of the sides that have the same relative position. These corresponding sides are proportional in similar triangles. Let's consider a simple scenario where we have two similar triangles
A'B'C' and ABC.
The sides of A'B'C' are a', b', and c'.The sides of ABC are a, b, and c.If these two triangles are similar, the ratio of their corresponding sides will be equal. Hence our ratios will look like this:
a'/a = b'/b = c'/c.
This is the ratio of corresponding sides for the similar triangles. To reduce the ratio to the lowest term, you simply divide each term by the greatest common factor of all the terms.
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To find the ratio of corresponding sides of similar triangles, identify the corresponding sides and write a ratio comparing the lengths of these sides. For example, if side AB in triangle ABC corresponds to side DE in triangle DEF with lengths 3 and 6 respectively, the ratio is 3/6 or 1/2 when reduced.
Explanation:To write the ratio of corresponding sides for similar triangles, we must first identify which sides correspond in the two triangles. Usually, triangles are labeled so that corresponding sides have the same label (like "a", "b", etc.). Once you've identified which sides correspond, simply write a ratio comparing the length of one side in the first triangle to the length of the corresponding side in the second triangle. This could look something like a1/a2.
For example, if triangle ABC is similar to triangle DEF and side AB corresponds to DE with lengths 3 and 6, respectively, then the ratio of the corresponding sides is 3/6 or 1/2 in reduced form.
Understanding the ratio of corresponding sides is key to understanding the properties of similar triangles and can be extremely helpful in solving problems involving these figures not just in geometry, but in other branches of mathematics as well.
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Q #3. Graph the equation
In virginia, each automobile license plate consists of a single digit followed by three letters, followed by three digits. how many distinct license plates can be formed if there are no restrictions on the digits or letters?
The number of distinct license plates can be formed if there are no restrictions on the digits or letters is 175,760,000.
Given that, in Virginia, each automobile license plate consists of a single digit followed by three letters, followed by three digits.
What is the Permutations?Permutations are different ways of arranging objects in a definite order. It can also be expressed as the rearrangement of items in a linear order of an already ordered set.
There are 26 letters A to Z in the alphabet. There are 10 digits 0 to 9
Assuming repetition allowed.
Now, 10×26×26×26×10×10×10
= 175,760,000
Hence, the number of distinct license plates can be formed if there are no restrictions on the digits or letters is 175,760,000.
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Suppose that the dollar cost of producing x radios is c(x) = 400 + 20x - 0.2x 2. Find the marginal cost when 30 radios are produced.
The marginal cost when 30 radios are produced is $8.
Explanation:To find the marginal cost when 30 radios are produced, we need to differentiate the cost function with respect to x, which represents the number of radios produced. The cost function is given as c(x) = 400 + 20x - 0.2x^2. Differentiating c(x) with respect to x, we get c'(x) = 20 - 0.4x. Now, substitute x = 30 into c'(x) to find the marginal cost when 30 radios are produced. c'(30) = 20 - 0.4(30) = 20 - 12 = 8. Therefore, the marginal cost when 30 radios are produced is $8.
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The graph shows the function f(x).
Which value is closest to the average rate of change from x = 1 to x = 4?
A.−3.5
B.−2.3
C. −1.4
D .−0.3
Answer:
Option B is correct
-2.3
Step-by-step explanation:
Average rate of change(A(x)) for the function f(x) over the interval [a, b] is given by:
[tex]A(x) = \frac{f(b)-f(a)}{b-a}[/tex] ....[1]
We have to find the average rate of change from x = 1 to x = 4.
From the graph as shown below
For x = 1
f(1) = 3
and
For x = 4
f(4) = -3.9
Using [1] we have;
[tex]A(x) = \frac{f(4)-f(1)}{4-1}[/tex]
Substitute the given values we have;
[tex]A(x) = \frac{-3.9-3}{3}[/tex]
⇒[tex]A(x) = \frac{-6.9}{3}[/tex]
Simplify:
[tex]A(x) = -2.3[/tex]
Therefore, -2.3 value is closest to the average rate of change from x = 1 to x = 4
Resting heart rate was measured for a group of subjects; the subjects then drank 6 ounces of coffee. ten minutes later their heart rates were measured again. the change in heart rate followed a normal distribution, with mean increase of 7.3 beats per minute and a standard deviation of 11.1 beats per minute. let latex: y y denote the change in heart rate for a randomly selected person. find latex: \text{p}(y<10)
Calculate the probability that the change in heart rate is less than 10 beats per minute using z-scores and the standard normal distribution table.
The probability (P) that the change in heart rate (y) is less than 10 beats per minute is calculated by finding the z-score for 10, then using the z-table or a calculator to find the corresponding probability.
First, calculate the z-score: z = (10 - 7.3) / 11.1 = 0.2432. Next, find the probability by looking up this z-score in the standard normal distribution table, which corresponds to approximately 59.93%.
Therefore, the probability that the change in heart rate is less than 10 beats per minute is approximately 59.93%.
Then find the area bounded by the two graphs of y=2x^2−24x+42 and y=7x−2x^2
The mean of a curriculum committee is 34.8 years. A 15-year-old student representative is added to the committee. How does the student’s age affect the mean?
A.) The new mean age will be less than 34.8
B.) The new mean age will be greater than 34.8
C.) The new mean age will still be 34.8
D.) The new mean age will be 24.9
Answer:
The correct option is A. The new mean age will be less than 34.8
Step-by-step explanation:
The mean of a curriculum committee is 34.8 years
Now, A 15-year-old student representative is added to the committee
Number of students in the curriculum committee is not known so if a student of 15 year age is added into the society we cannot find the exact new mean of the curriculum committee.
But, As the 15 is added to the sum of all observations but only 1 is added to the number of observations
So, The mean of the curriculum committee will obviously decrease
Since, Th mean of the curriculum committee is 34.8
Therefore, the new mean of the curriculum committee will be less than 34.8
Hence, The correct option is A. The new mean age will be less than 34.8
he radius of a circle is 2 kilometers. What is the area of a sector bounded by a 45° arc?
The area of a sector bounded by a 45° arc in a circle with a radius of 2 kilometers is π/² square kilometers.
To calculate the area of a sector bounded by a 45° arc in a circle with a radius of 2 kilometers, we will use the formula for the area of a sector, which is (θ/360) × π × r², where θ is the central angle in degrees and r is the radius of the circle. The central angle for our sector is 45° and the radius r is given as 2 km.
Plugging these values into the formula, we have:
Area of sector = (45/360) × π × (2²) = (1/8) × π × 4 = (1/2) × π = π/2 km².
Therefore, the area of the sector bounded by a 45° arc in a circle with a radius of 2 kilometers is π/²square kilometers.
A hospital needs a circular landing pad for their helicopter. The landing pad should be 5000 square feet. The contractor calculated that the pad would need to be about 40 feet across. Formulas Is the contractor correct? If not, what error did the contractor make?
Yes, the answer is correct.
No, he forgot to multiply by pi (3.14).
No, he forgot to multiply the radius by 2.
No, he used the formula for circumference.
Why is a/0 not defined?
Final answer:
Division by zero is undefined because a/0 would imply an infinite value not included in the real numbers, and 0/0 is an indeterminate form lacking unique value, requiring advanced techniques in calculus for evaluation.
Explanation:
The concept of division by zero, specifically when discussing expressions like a/0 and 0/0, is a fundamental aspect of mathematics that leads to the undefined nature of these operations. In the case of 1/0, this division is not defined within the real number system, as a non-zero number divided by zero would imply an infinite value, which is outside the bounds of real numbers.
Conversely, the expression 0/0 is an example of an indeterminate form because it doesn't present enough information to deduce a unique value for the division, as zero divided by zero could represent any number.
Indeterminate forms such as these necessitate a more nuanced approach, particularly in calculus where the evaluation of limits often brings these expressions into play. In some contexts, sophisticated mathematical techniques must be employed to determine the behavior of functions as they approach these forms.
In science class, students are learning about organic compounds. An acetic acid molecule is made of 2 carbon atoms, 2 oxygen atoms and 4 hydrogen atoms. What is the ratio of carbon atoms and hydrogen atoms
Acetic acid, with the molecular formula C2H4O2, has a carbon to hydrogen atom ratio of 2:4, which simplifies to a 1:2 ratio.
The student's question pertains to the ratio of carbon atoms to hydrogen atoms in an acetic acid molecule. Acetic acid, which is also known as vinegar, has the molecular formula C2H4O2. The ratio of carbon to hydrogen atoms can be found by considering the number of carbon atoms and the number of hydrogen atoms in the formula. There are 2 carbon atoms and 4 hydrogen atoms, resulting in a 2:4 ratio. However, this ratio can be simplified by dividing both numbers by the greatest common divisor, which is 2. After simplifying, the ratio of carbon to hydrogen atoms becomes 1:2.
(3) (-4) + (3) (4) -1
Sam left his school at 3:05. He walked at a speed of 3.2 mph. 15 minutes later, Al started running after him, and he caught up with Sam 10 minutes later. What was Al’s speed? Answer:
What were the total earnings of all five of these movies in the given week?
Movie
Earnings
Average Ticket Price
A
$26,088,808.74
$8.35
B
$60,394,938.12
$9.72
C
$23,659,617.52
$8.12
D
$34,311,887.98
$7.57
E
$10,505,611.08
$8.46
Answer:
$154,960,863.44
Step-by-step explanation:
Add the 5 earnings numbers using a suitable calculator.
_____
In this case, a "suitable calculator" is one that will display numbers of 11 digits or more. Apparently the one at the Google search box is up to the task.
The total earnings of all five movies in the given week were approximately $155,950,863.44.
To find the total earnings of all five movies in the given week, you can simply add up their individual earnings:
Total Earnings = Earnings of Movie A + Earnings of Movie B + Earnings of Movie C + Earnings of Movie D + Earnings of Movie E
Total Earnings = $26,088,808.74 + $60,394,938.12 + $23,659,617.52 + $34,311,887.98 + $10,505,611.08
Now, calculate the sum:
Total Earnings = $155,950,863.44
So, the total earnings of all five movies in the given week were approximately $155,950,863.44.
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Which figure has all sides of equal measure but not necessarily all angles of equal measure?
The correct figure which has all sides of equal measure but not necessarily all angles of equal measure are ''Rhombus'' and ''parallelogram''.
What is mean by Rectangle?A rectangle is a two dimension figure with 4 sides, 4 corners and 4 right angles. The opposite sides of the rectangle are equal and parallel to each other.
Given that;
To find figure which has all sides of equal measure but not necessarily all angles of equal measure.
Now, We know that;
In a Parallelogram, A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides of a parallelogram are equal in length.
And, In a Rhombus, A rhombus is a quadrilateral with both pairs of opposite sides parallel and all sides the same length,
Thus, The correct figure which has all sides of equal measure but not necessarily all angles of equal measure are Rhombus and parallelogram.
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Identify the type of conic section whose equation is given. x2 = y + 3
Given: PSTR is a parallelogram m∠T:m∠R=1:3, RD ⊥ PS , RM ⊥ ST Find: m∠DRM
Answer:
m∠DRM = 45°
Step-by-step explanation:
∵ PSTR is a parallelogram
∴ TS // RP ⇒ opposite sides
∴ m∠T + m∠R = 180° ⇒ (1) (interior supplementary angles)
∵ m∠T : m∠R = 1 : 3
∴ m∠R = 3 m∠T ⇒ (2)
- Substitute (2) in (1)
∴ m∠T + 3 m∠T = 180
∴ 4 m∠T = 180
∴ m∠T = 180 ÷ 4 = 45°
∴ m∠R = 3 × 45 = 135°
∵ m∠R = m∠S ⇒ opposite angles in a parallelogram
∴ m∠S = 135°
∵ RD ⊥ PS
∴ m∠RDS = 90°
∵ RM ⊥ ST
∴ m∠RMS = 90°
- In quadrilateral RMSD
∵ m∠S = 135°
∵ m∠RDS = 90°
∵ m∠RMS = 90°
∵ The sum of measure of the angles of RMSD = 360°
∴ m∠DRM = 360 - ( 135 + 90 + 90) = 45°
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PLEEEEEEEEEEEEEEAAAAAASSSSSSSSSSSSSEEEEEEEEEEEEEEEEE HELP!!!
The figure below is an oblique triangular prism. The expression below represents the volume of the figure written in standard form.
What are the missing values? axb + cx a = b = c =
Suppose a city has 810 high-rise buildings, and 29 of these buildings have rooftop gardens. Find the percentage of high-rise buildings with rooftop gardens in this city. Round your answer to the nearest tenth of a percent.
To find the percentage of high-rise buildings with rooftop gardens, divide the number of buildings with gardens (29) by the total number of buildings (810), and then multiply by 100. Round the final result to the nearest tenth, which is approximately 3.6%.
To calculate the percentage of high-rise buildings with rooftop gardens, we use the formula:
Percentage = (Part / Whole) imes 100
Where the Part is the number of buildings with rooftop gardens, and the Whole is the total number of high-rise buildings.
Substituting the given values:
Percentage = (29 / 810) times 100
Carrying out the division first gives us:
Percentage ≈ 0.035802469 times 100
Finally, multiplying by 100 to find the percentage, we get:
Percentage ≈ 3.58
After rounding to the nearest tenth of a percent, we obtain:
Percentage ≈ 3.6%
3.6% of the high-rise buildings in the city have rooftop gardens.
Figure BMHF is rotated how many degrees clockwise?
The center is (4,5) and a point on the circle is (0,8)
A rectangular room measures 12 ft by 17 fr.Find the cost of installing a wall paper border around the room if the border costs $0.64 cents per foot
A tangent- tangent intercept two arcs that measure 145 degrees and 211 degrees what is the measure of the length of the tangent-tangent angle?
The answer to this question is A: 31
Hannah made 0.7 of her free throws in a basketball game. Abra made 9/10 of her free throws. Dena made 3/4 of her free throws.Who was the best shooter? explain
Abra made 9/10 of her free throws which are equal to 90%, so Abra is the best shooter.
What is the percentage?A percentage is a number or ratio that can be expressed as a fraction of 100. If we have to calculate the percent of a number, divide the number by the whole and multiply by 100.
Hannah made 0.7 of her free throws in a basketball game.
[tex]\rm = 0.7 \times 100\\\\=70 \ percent[/tex]
Hannah made 0.7 of her free throws in a basketball game which is equal to 70%.
Abra made 9/10 of her free throws.
[tex]\rm =\dfrac{9}{10}\times 100\\\\= 9\times 10\\\\=90 \ percent[/tex]
Abra made 9/10 of her free throws which are equal to 90%.
Dena made 3/4 of her free throws.
[tex]\rm =\dfrac{3}{4} \times 100\\\\=3\times 25\\\\=75 \ percent[/tex]
Dena made 3/4 of her free throws which are equal to 75%.
Hence, Abra made 9/10 of her free throws which are equal to 90%, so Abra is the best shooter.
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the two figures are congruent find the measure of the requested side or angle
The length of AB is approximately 14.68 which is option B. 15.
To find the length of side AB in the congruent figures where AC is 6, angle C is 119 degrees, AB is 15, angle B is 22 degrees, and BC is 12, we can use the Law of Cosines.
The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(C) \][/tex]
where:
- c is the side opposite the angle C,
- a and b are the other two sides,
- c is the angle opposite side c.
In this case, we have AC as side a, BC as side b, and AB as side c. We also know that angle C is 119 degrees.
[tex]\[ AB^2 = AC^2 + BC^2 - 2 \cdot AC \cdot BC \cdot \cos(C) \][/tex]
Substitute the given values:
[tex]\[ AB^2 = 6^2 + 12^2 - 2 \cdot 6 \cdot 12 \cdot \cos(119^\circ) \][/tex]
Now, calculate the expression to find the length of AB.
[tex]\[ AB^2 = 36 + 144 - 2 \cdot 6 \cdot 12 \cdot \left(\cos(119^\circ)\right) \]\\AB^2 = 36 + 144 - 2 \cdot 6 \cdot 12 \cdot \left(-0.492403\right) \]\\AB^2 = 36 + 144 + 35.4234 \]\\AB^2 = 215.4234 \][/tex]
Now, take the square root of both sides to find AB:
[tex]\[ AB = \sqrt{215.4234} \approx 14.68 \][/tex]
So, the length of AB is approximately 14.68.
One month Mai rented
6 movies and 2 video games for a total of
$36. The next month she rented 3
movies and 5 video games for a total of $39. Find the rental cost for each movie and each video game.
Which polynomial function has a leading coefficient of 1, roots –3 and 8 with multiplicity 1, and root 4 with multiplicity 2?
f(x) = 2(x + 3)(x + 4)(x – 3)
f(x) = 2(x – 8)(x – 4)(x + 3)
f(x) = (x + 8)(x + 4)(x + 4)(x – 3)
f(x) = (x – 8)(x – 4)(x – 4)(x + 3)
Answer:
f(x) = (x – 8)(x – 4)(x – 4)(x + 3)
Step-by-step explanation:
A polynomial function with roots [tex]x_{1}, x_{2}, ..., x_{n}[/tex] has the following format:
[tex]f(x) = a(x - x_{1})(x - x_{2})...(x - x_{n})[/tex]
In which a is the leading coefficient.
In this problem, we have that:
Leading coefficient 1, so [tex]a = 1[/tex]
roots -3 and 8 with multiplicity 1, so [tex](x + 3)(x - 8)[/tex].
root 4 with multiplicity 2, so [tex](x - 4)^{2} = (x - 4)(x - 4)[/tex]
So the correct answer is:
f(x) = (x – 8)(x – 4)(x – 4)(x + 3)
Which side of XYZ is the longest ?? Please Help
Answer:
XY is the longest side in the given ΔXYZ.
Step-by-step explanation:
We are given the following information in the question:
We have to find the longest side in the given triangle ΔXYZ.
The three angles of the triangle are:
[tex]\angle X = 62^\circ\\\angle Y = 55^\circ\\\angle Z = 63^\circ\\[/tex]
We know that in a triangle the side opposite to smallest triangle is smallest and the side opposite to largest angle is longest in length.
Angle Z in the given triangle is the largest angle and therefore, the side opposite to this angle is the longest side of the triangle.
Hence, XY is the longest side in the given ΔXYZ.
Use part 1 of the fundamental theorem of calculus to find the derivative of the function. y = 5 u3 1 + u2 du 4 − 3x
Using part 1 of the fundamental theorem of calculus to find the derivative of the function. The derivative of the given function is:
[tex]\mathbf{\dfrac{dy}{dx} =3\Big (\dfrac{(4-3x)^3}{1+(4-3x)^2}\ \Big) \ }[/tex]
Consider the given function:
[tex]\mathbf{y = \int^5_{4-3x} \ \dfrac{u^3}{1+u^2}\ du}[/tex]
The objective is to find [tex]\mathbf{\dfrac{dy}{dx}}[/tex] by using the fundamental theorem of calculus.
Suppose v = 4 - 3x; Then dv = -3dx[tex]\mathbf{\dfrac{dv}{dx}= -3}[/tex]Using chain rule:
[tex]\mathbf{\dfrac{dy}{dx} = \dfrac{dy}{dv}\times \dfrac{dv}{dx}}[/tex]
[tex]\mathbf{ =\dfrac{d}{dv}\Big (\int^5_{4-3x} \dfrac{u^3}{1+u^2}\ du\Big) \dfrac{dv}{dx}}}[/tex]
[tex]\mathbf{ =\dfrac{d}{dv}\Big (\int^5_{v} \dfrac{u^3}{1+u^2}\ du\Big) \dfrac{dv}{dx} \ \ \ \ \ since \ v \ = 4 - 3x} }[/tex]
[tex]\mathbf{ =-\dfrac{d}{dv}\Big (\int^v_{5} \dfrac{u^3}{1+u^2}\ du\Big) \dfrac{dv}{dx} }[/tex]
[tex]\mathbf{ =-\dfrac{d}{dv}\Big (\int^v_{5} \dfrac{u^3}{1+u^2}\ du\Big) \ (-3)}[/tex]
[tex]\mathbf{ =3\dfrac{d}{dv}\Big (\int^v_{5} \dfrac{u^3}{1+u^2}\ du\Big) \ }[/tex]
From the fundamental theorem of calculus;
[tex]\mathbf{\dfrac{d}{dx} \Big( \int^x_1 \ g(t) dt \Big) = g(x)}[/tex]
∴
[tex]\mathbf{ =3\dfrac{d}{dv}\Big (\int^v_{5} \dfrac{u^3}{1+u^2}\ du\Big) \ }[/tex] will be:
[tex]\mathbf{ =3\times \Big (\dfrac{(4-3x)^3}{1+(4-3x)^2}\ \Big) \ }[/tex]
∴
[tex]\mathbf{\dfrac{dy}{dx} =3\Big (\dfrac{(4-3x)^3}{1+(4-3x)^2}\ \Big) \ }[/tex]
Therefore, we can conclude that the derivative of [tex]\mathbf{y = \int^5_{4-3x} \ \dfrac{u^3}{1+u^2}\ du}[/tex]
using the fundamental theorem of calculus is [tex]\mathbf{\dfrac{dy}{dx} =3\Big (\dfrac{(4-3x)^3}{1+(4-3x)^2}\ \Big) \ }[/tex]
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