Use two different methods to find an explain the formula for the area of a trapezoid that has parallel sides of length a and B and height H. Consider these ideas or others: combine two copies of a trapezoid to make a parallelogram. Subdivide the trapezoid into two triangles one with base b and one with base a.
Answers
Answer:
Formula of Trapezoid:
A = (a + b) × h / 2
The formula can be derived in different ways. for now, we have discussed two ways:
1. By using the formula of a triangle
2. By dividing into different sections
Step-by-step explanation:
1. By using the formula of a triangle
One of the ways to explain a formula for an area of a trapezoid using a formula for a triangle can be as follows.
Assume a trapezoid PQRS with lower base SR and upper base PQ (they are parallel) and sides PS and QR.
The image is attached below.
Connect vertices P and R with a diagonal.
Consider triangle ΔPQR as having a base PQ and an altitude from vertex R down to point M on base PQ (RM⊥PQ).
Its area is
S1=[tex]\frac{1}{2} *PQ*RM[/tex]
Consider triangle ΔPRS as having a base SR and an altitude from vertex P up to point N on-base SR (PN⊥SR).
Its area is
S2=[tex]\frac{1}{2} *SR*PN[/tex]
Altitudes RM and PN are equal and constitute the distance between two parallel bases PQ and SR.
They both are equal to the altitude of the trapezoid h.
Therefore, we can represent areas of our two triangles as
S1=[tex]\frac{1}{2}*PQ*h[/tex]
S2=[tex]\frac{1}{2}*SR*h[/tex]
Adding them together, we get the area of the whole trapezoid:
S=S1+S2=[tex]\frac{1}{2} (PQ+SR)h,[/tex]
which is usually represented in words as "half-sum of the bases times the altitude".
2. By dividing into different sections
Trapezoid PQRS is shown below, with PQ parallel to RS.
Figure 1 - Trapezoid PQRS with PQ parallel to RS(image is attached below.)
We are going to derive the area of a trapezoid by dividing it into different sections.
If we drop another line from Q, then we will have two altitudes namely PT and QU.
Figure 2 - Trapezoid PQRS divided into two triangles and a rectangle. (image is attached below.)
From Figure 2, it is clear that Area of PQRS = Area of PST + Area of PQUT + Area of QRU. We have learned that the area of a triangle is the product of its base and altitude divided by 2, and the area of a rectangle is the product of its length and width. Hence, we can easily compute the area of PQRS. It is clear that
=> [tex]A_{PQRS} = (\frac{ah}{2}) + b_{1}h + \frac{ch}{2}[/tex]
Simplifying, we have
=>[tex]A= \frac{ah+2b_{1+C} }{2}[/tex]
Factoring we have,
=> [tex]A_{PQRS} = (a+ 2b_{1} + c)\frac{h}{2} \\= > {(a+ b_{1} + c) + b_{1} }\frac{h}{2}[/tex]
But, [tex]a+ b_{1} + c[/tex] is equal to [tex]b_{2}[/tex], the longer base of our trapezoid.
Hence, [tex]A_{PQRS}= (b_{1} + b_{2} )\frac{h}{2}[/tex]
We have discussed two ways by which we can derive area of a trapezoid.
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Related Questions
A rectangular floor of area 21.6 m2 is going to be tiled. Each tile is rectangular, and has an area of 288 cm2. An exact number of tiles can be put into the space. How many tiles will be needed?
Answers
Answer:
Area of Floor = 21.6.m²
Area of tile = 288 cm² = 0.0288 m²
No. of tiles = Area of floor/ area of tile
= 21.6/0.0288
= 750
750 tiles can be put into the spaceDJ Erin is making a playlist for work; he is trying to decide what 10 songs to play and in what order they should be played.
Step 2 of 2 : If he has his choices narrowed down to 3 country, 6 pop, 7 disco, and 3 hip-hop songs, and he wants to play all 7 disco songs, how many different playlists are possible? Express your answer in scientific notation rounding to the hundredths place.
Answers
Using the combination and the arrangements formula, it is found that the number of possible playlists is given by:
[tex]7.98 \times 10^8[/tex]
What is the combination formula?[tex]C_{n,x}[/tex] is the number of different combinations of x objects from a set of n elements, given by:
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
In this problem, we have that:
7 disco songs are taken from a set of 7.3 remaining songs are taken from a set of 3 + 6 + 3 = 12 songs.Hence, without considering the order, the number of playlists is given by:
[tex]C_{7,7}C_{12,3} = \frac{7!}{7!0!} \times \frac{12!}{3!9!} = 220[/tex]
What is the arrangements formula?The number of possible arrangements of n elements is given by the factorial of n, that is:
[tex]A_n = n![/tex]
In this problem, the 10 musics are arranged, hence the number of playlists, considering the order, is given by:
[tex]n = 10! \times 220 = 798336000 = 7.98 \times 10^8[/tex]
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The amount left from you net income after your fixed expenses have been paid.
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The profit is the amount left from you net income after your fixed expenses have been paid.
The amount earned by an individual or corporation after costs, allowances, and taxes is referred to as net income. Net income in company is what remains after all costs, such as salaries and wages, the cost of goods or raw materials, and taxes, have been paid.
Profit is a financial gain, specifically the sum obtained over that which was used to acquire, maintain, or produce the good or service.
Profit is the term used to describe the financial gain experienced when the revenue from a commercial activity exceeds the costs, costs, and taxes associated with maintaining that activity.
Hence the profit is the amount left from you net income after your fixed expenses have been paid.
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if x²+X+1=0 then what is the value of x²⁰¹⁵ + x²⁰¹³ + x
Answers
Since [tex]x^2 + x + 1 = 0[/tex] means [tex]x^2+x = -1[/tex], we have for any [tex]n[/tex] the recursive relation
[tex]x^n (x^2 + x) = x^{n+2} + x^{n+1} = -x^n \implies x^{n+2} = -x^{n+1} - x^n[/tex]
Let [tex]f(x)=x^{2015}+x^{2013}+x[/tex]. Substitution using the recursive rule generates an alternating power-reduction pattern:
[tex]f(x) = \left(-x^{2014} - x^{2013}\right) + x^{2013} + x = -x^{2014} + x[/tex]
[tex]f(x) = -\left(-x^{2013} - x^{2012}\right) + x = x^{2013} + x^{2012} + x[/tex]
[tex]f(x) = \left(-x^{2012} - x^{2011}\right) + x^{2012} + x = -x^{2011} + x[/tex]
[tex]f(x) = -\left(-x^{2010} - x^{2009}\right) + x = x^{2010} + x^{2009} + x[/tex]
[tex]f(x) = \left(-x^{2009} - x^{2008}\right) + x^{2009} + x = -x^{2008} + x[/tex]
[tex]f(x) = -\left(-x^{2007} - x^{2006}\right) + x = x^{2007} + x^{2006} + x[/tex]
and so on.
Notice that in the equivalent forms of [tex]f(x)[/tex] involving 3 terms, the largest power of [tex]x[/tex] is a multiple of 3, and the next largest power is 1 less. This means after so many iterations of substitutions, we would end up with
[tex]f(x) = x^3 + x^2 + x[/tex]
Then by factorizing, the expression of interest reduces to
[tex]f(x) = x (x^2 + x + 1) = x \times 0 = \boxed{0}[/tex]
Steve bought a new car for $22,000 but paid 93% of the list price. How much was the list price
Answers
Answer:
approximately $23,655.91 (rounded to the nearest hundredths place)
Step-by-step explanation:
[tex]22,000=0.93x\\x=23,655.91[/tex]
Step my step explanation:
In this problem, we are trying to find 100 percent of the 93% Steve paid for his new car. First we will find the 7% Steve has not paid yet. If Steve paid $22,000, that would be 93% of the list price.
1.) Multiply 22,000 by .07 (or 7 percent)
2.) Add the 7 percent ($1540) to $22,000 (The amount he already paid)
3.) Get the result 23,540!
Let me know if you have any questions :$
How do I do this? The question is in the picture.
Answers
Answer:
Step-by-step explanation:
simplify (8x2y3)/(2x4y)
Answers
Answer:
1) 8x/2x = 4
2) 2y/4y = 1/2
3) (4 x 1/2 x 3) = 6
4) /2
solutions= 6/2=3
Phil Santos owns an independent mobile phone store. He has a wonderful personality, is very knowledge about the products he sells and people flock to his store. Phil would like to start selling online so he can reach more customers.
He's talked with website developers and the firm he decided to go with will charge him $9500 to build the site. Phil wondered why the site was so expensive and explored the idea of building it himself. He found out it's complicated to build a site that can take payments so he decided to be safe rather than sorry and have it built by experts. They also gave him an estimate of $5300 a year to maintain his site. This will allow him to update the site constantly and add new pictures and videos as new products and promotions are introduced over the course of the year.
A marketing research firm found that he should expect about 133 sales a month averaging about $290 profit per sale. Phil was thrilled with this news and decided to spend $15,000 to add a new room to his store to be used as a shipping area.
He also wanted to make his sales literature and in store promotional materials such as banners, and promotional give-a-ways consistent with the look of the website. Phil has worked with a graphic design class at the local community college and knows it will cost $2300 a year to have the materials designed and printed.
What is the cost to acquire each new customer? Show your full calculation.
Does it make sense for Phil to sell his products online?
Answers
Answer:
It makes sense for Phil to sell his products online.
Step-by-step explanation:
X: 4x^2+16 -20=0
What is x
Answers
X=5
Answer:
[tex]\boxed{\sf x = 1, \ x = -5}[/tex]
Explanation:
[tex]\rightarrow \sf 4x^2 + 16x - 20 = 0[/tex]
[tex]\rightarrow \sf 4(x^2 + 4x - 5) = 0[/tex]
[tex]\rightarrow \sf x^2 + 4x - 5 = 0[/tex]
[tex]\rightarrow \sf x^2 + 5x -x- 5 = 0[/tex]
[tex]\rightarrow \sf x(x + 5) -1(x+ 5) = 0[/tex]
[tex]\rightarrow \sf (x-1)(x+ 5) = 0[/tex]
[tex]\rightarrow \sf x = 1, \ x = -5[/tex]
find the area of the shaded portion in each of the following
Answers
Answer:
30.5 cm^2
Step-by-step explanation:
General outlineFind the area of the full circleFind the area of the rectangleFind the difference (full circle area - rectangle area)Step 1. Find the area of the full circleRecall the formula for the area of a circle: [tex]A_{circle}=\pi r^2[/tex] where [tex]\pi \approx 3.14[/tex], and "r" is the radius of the circle (distance from center to any point on the edge).
From the diagram, the radius is 5cm.
[tex]A_{circle}=\pi r^2[/tex]
[tex]A_{circle}=(3.14)(5[cm])^2[/tex]
[tex]A_{circle}=78.5[cm^2][/tex]
Step 2. Find the area of the rectangleRecall the formula for the area of a rectangle: [tex]A_{rectangle}=bh[/tex] where "b" is the base of the rectangle (length of bottom or length of top), "h" is the height of the rectangle (distance from bottom to top).
From the diagram, the base is 8cm, and the height is 6cm.
[tex]A_{rectangle}=bh[/tex]
[tex]A_{rectangle}=(8[cm])(6[cm])[/tex]
[tex]A_{rectangle}=48[cm^2][/tex]
Step 3. Find the difference (full circle area - rectangle area)Having found the areas for each of the first parts, we can find the difference to find the shaded area:
[tex]A_{shaded}=A_{circle}-A_{rectangle}[/tex]
[tex]A_{shaded}=(78.5[cm^2])-(48[cm^2])[/tex]
[tex]A_{shaded}=30.5[cm^2][/tex]
[tex]4x {}^{2} + 20x + 25[/tex]
using appropriate identity factories the following
Answers
[tex] \sf{\qquad\qquad\huge\underline{{\sf Answer}}} [/tex]
Let's Solve ~
[tex]\qquad \sf \dashrightarrow \: 4 {x}^{2} + 20x + 25[/tex]
[tex]\qquad \sf \dashrightarrow \: 4 {x}^{2} + 10x + 10x + 25[/tex]
[tex]\qquad \sf \dashrightarrow \: 2x(2x + 5) + 5(2x + 5)[/tex]
[tex]\qquad \sf \dashrightarrow \: (2x + 5) (2x + 5)[/tex]
[tex]\qquad \sf \dashrightarrow \: (2x + 5) {}^{2} [/tex]
or
[tex] \sf{\qquad \sf \dashrightarrow \: 4x² + 20x + 25 } [/tex]
[tex] \sf{\qquad \sf \dashrightarrow \: (2x)² + (2 \sdot 2x \sdot 5) + (5)²} [/tex]
[ it's similar to expression - a² + 2ab + b² that is equal to (a + b)² ]
so, let's use this identity here to factorise :
[tex] \sf{\qquad \sf \dashrightarrow \: (2x + 5)²} [/tex]
I hope it was helpful ~
help me please ! due soon!
Answers
Answer:
Step-by-step explanation:
Comment
An x intercept occurs when y = 0. So the table shows three of them.
Answer
There are 3 x values when y = 0. They are
x = - 1x = 2x = 3Answer: A. [tex](-1, 0)[/tex]
Step-by-step explanation:
The x-intercept is every point where y=0. To see this, we should look at the table's f(x) section to find any zeroes.
According to the table and this knowledge, one x-intercept of this continuous function is [tex](-1, 0)[/tex].
I hope this helps! Pls mark brainliest!! :)
4.5x + 9 -1/2x = -3.75 +5 1/2 + 5.25
Answers
Answer:
x = -1/2 = -0.5
Step-by-step explanation:
4.5x + 9 -1/2x = -3.75 +5 1/2 + 5.25
⇔ (4 1/2 - 1/2)x + 9 = (5 1/2 + 5 1/4) - 3 3/4
⇔ 4x + 9 = 10 3/4 - 3 3/4
⇔ 4x + 9 = 10 - 3
⇔ 4x + 9 = 7
⇔ 4x = 7 - 9
⇔ 4x = -2
⇔ x = -2/4
⇔ x = -1/2
⇔ x = -0.5
Find the radius of a circle with area 345 square centimetres
For 35 points need asap in 10 MINS!!!! :)
Answers
5.9
Step-by-step explanation:
345√ is 18.6=3.14*r2 cuz formula
Is πr2
Step-by-Step Solution:
Radius (r) = ?
Area (A) = πr² = 345 sq. cm
Therefore,
πr² = 345
3.14 * r² = 345
r² = 345/3.14
r² = 109.87
r = √109.87
=> r = 10.48
Radius (r) = 10.48 cm
Which property of real numbers IS shown below?
3 + ((-5) + 6) = (3 + (-5)) + 6
Answers
The associative property of real numbers is shown in the number expression 3 + ((-5) + 6) = (3 + (-5)) + 6
What is an arithmetic operation?It is defined as the operation in which we do the addition of numbers, subtraction, multiplication, and division. It has a basic four operators that is +, -, ×, and ÷.
We have a number expression:
= 3 + ((-5) + 6) = (3 + (-5)) + 6
As we know in the associative property:
a + (b + c) = (a + b) + c
= (3 + (-5)) + 6
Thus, the associative property of real numbers is shown in the number expression 3 + ((-5) + 6) = (3 + (-5)) + 6
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e:f=3:7
f:g=2:3
Work out e:g in simplest form
Answers
The value of e:g is 2:7.
what is Ratio?A ratio indicates how many times one number contains another.
Given:
e:f=3:7
f:g=2:3
Now take LCM of 2 and 7 i.e., 14.
e/f= 3/7= 3*2/7*2= 6/14
and, f/g= 2/3= 2*7/3*7= 14/21
So,
e/g= e/f *f /g
= 6/14*14/21
= 6/21
e:g = 2:7
Hence, the value of e:g is 2:7.
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14 A ladder 10m long leans against a wall and make 45" with the ground. What is the height of the wall? 14 A ladder 10m long leans against a wall and make 45 " with the ground . What is the height of the wall ?
Answers
Answer:
A 10m ladder is leaning against a wall. The ladder makes an angle of 78o with the ground. How high up the wall does the ladder reach? Please include a diagram in the answer.
Th function you need is the sine function. The sine of 78° is about 0.978. That is the ratio of the opposite side (wall) over the hypotenuse (ladder) of the right triangle in this problem.
10 m · sin 78° = 10 m · 0.978 = 9.78 meters.
The ladder reaches 9.78 meters up on the wall.
Step-by-step explanation:
Chico, California, hosts the annual Silver Dollar Fair.
In 2016, contestants competed in the Inaugural World Silver Dollar
Pancake Eating Championship, where they had 8 minutes to eat as
many one-ounce silver dollar pancakes as possible.
Answers
Answer: what is the question
Step-by-step explanation:
Please help with the attached photo.
Answers
The solution to the equation r(1 - 2cosФ) = 1 is given as x² + y² - 4x√(x² + y²) + 4x² = 1
What is an equation?An equation is an expression that shows the relationship between two or more variables and numbers.
In polar form:
r = √(x² + y²) and cosФ = x / √(x² + y²)
Hence:
r(1 - 2cosФ) = 1
√(x² + y²) [1 - 2(x / √(x² + y²))] = 1
√(x² + y²) - 2x = 1
Take square of both sides:
x² + y² - 4x√(x² + y²) + 4x² = 1
The solution to the equation r(1 - 2cosФ) = 1 is given as x² + y² - 4x√(x² + y²) + 4x² = 1
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Write a slope-intercept equation for a line that passes through (-2,1) and (3,-14).
Answers
Suppose that an examination consists of 10 questions with the choice A, B, C, and D. Assume that a student has no knowledge of the subject matter. Then what is the probability that: Guessing at most two correct answers? Guessing at least seven correct answers? Which type of probability distribution it has? Why? average five smokers pass a crain traat B. C. Suppose that an examination consists of 10 questions with the choice A , B , C , and D. Assume that a student has no knowledge of the subject matter . Then what is the probability that : Guessing at most two correct answers ? Guessing at least seven correct answers ? Which type of probability distribution it has ? Why ?
Answers
The probability of getting two correct answers will be 0.0625.
How to calculate the probability?The probability of getting two correct answers will be:
Probability of getting the correct answer will be:
= 1/4 = 0.25
The probability of getting two correct answers will be:
= 0.25 × 0.25
= 0.0625
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Find f(-4) for f(x)=9(3)^x
Answers
Answer:
[tex] f(-4) = \dfrac{1}{9} [/tex]
Step-by-step explanation:
[tex] f(x) = 9(3)^x [/tex]
[tex] f(-4) = 9(3)^{-4} [/tex]
[tex] f(-4) = 3^2 \cdot 3^{-4} [/tex]
[tex] f(-4) = 3^{-2} [/tex]
[tex] f(-4) = \dfrac{1}{9} [/tex]
Answer:
1/9
-------------
i needs a help please
Answers
Answer:
Step-by-step explanation:
Givens
x intercept =
(1,0) (5,0)The vertex is at 3, 4
The vertex is upside down.
Solution
(x - 1) and (x - 5) produce the two roots.
There must be 1 minus somewhere so that the quadratic goes upside down.
Answer
y = - (x -1)(x - 5)
s x – 5 a factor of the function f(x) = x3 – 5x2 + 7x – 35? Use the Remainder Theorem to justify your answer.
Answers
By using the remainder theorem we found that ( x - 5 ) is the factor of function f(x) = x³ – 5x² + 7x – 35.
What is factorization?Factorization is defined as splitting the polynomial into its most simplified form. This factor satisfies the polynomials from which it is extracted.
The factorization by using the Remainder theorem.
x - 5 ) x³ – 5x² + 7x – 35.( x ² + 7
x³ - 5x²
______________
00 7x - 35
7x - 35
-------------------------
00
Here the function is completely divided by the factor x - 5 therefore
by using the remainder theorem we found that ( x - 5 ) is the factor of function f(x) = x³ – 5x² + 7x – 35.
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Is the graph increasing, decreasing, or constant?
Answers
Answer:
jxdhjsssdddddsssssdd
Answer:
increasing
Step-by-step explanation:
look left to right, line goes up
Two regular polygons X and Y are such that the number of sides of X is
3 more than the number of sides of Y. If the sum of the exterior angles
of X and Y is 117°, how many sides has .X?
Answers
Based on the indices of the regular polygons, the number of sides that X has is 8. See the computation below.
What is the analysis for the above solution?Assuming that X has G number of sides; and
Y has H number of sides, this means that:
G-H = 3 ...........1
Recall that the formula for the exterior angles is:
360/n; where n is number of sides
X = 360/G
Y = 360/H
(360/G) + (360/H) = 117°.......2
Solving 1 and 2 simultaneously, we obtain:
G= 8 and
H = 5
Thus, X has 8 sides.
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Find the area of the rectangle.
A rectangle is shown. The left side is labeled with the expression 8 x. The top side is labeled with the expression 7 x plus 1. (1 point)
15x + 1
56x2 + 8x
56x + 8
15x2 + 9x
Answers
Answer:
56x² + 8x
(second option listed)
Step-by-step explanation:
area: length × width
length: 7x + 1
width: 8x
(8x)(7x + 1)
{distribute 8x}:
(8x)(7x): 8x × 7x = 8 · 7 · x · x = 56 · x · x = 56x²
(8x)(1) = 8x
area = 56x² + 8x
so, the area of this rectangle is 56x² + 8x
Answer:
56x² + 8x
[answer B]
The reverse of the number 129 is 921, and these add to 1050, which is divisible by 30.
How many three-digit numbers have the property that, when added to their reverse,
the sum is divisible by 30?
Answers
Answer:
Step-by-step explanation:
Write the number as [tex]abc[/tex] and use the divisibility rules for 10 and 3. From the rule for 10, we know that the last digit of [tex]abc+cba[/tex] is [tex]o[/tex], hence [tex]a+c=10[/tex]. Thus the options are [tex]a=1[/tex] & [tex]c=9[/tex], [tex]a=2[/tex], & [tex]c=8[/tex], etc.
From the rule for 3, namely that the digit sum must be divisible by 3, we find that [tex]2a+2b+2c=20+2b[/tex] is divisible by [tex]3[/tex]. By hand it's easy to check that [tex]b=2,5,8[/tex] are the only options that work.
So there are 9 choices for [tex]a[/tex] and [tex]c[/tex] and [tex]3[/tex] for [tex]b[/tex] , giving 27 total.
Consider figures 1 and 2 shown in the coordinate plane. Figure 1 has been transformed to produce figure 2.
Answers
Answer:
0 and -6
(0,-6)
Step-by-step explanation:
Can't really explain, but this is a guess on the actual question.
How many degrees does Earth turn in one hour?
Answers
Answer:
15 degrees
Step-by-step explanation:
Answer:
The earth turns 15 degrees in one hour. Hope this helps!
Step-by-step explanation:
