Answer:
7720 mm²
Step-by-step explanation:
area = 2 × 80 mm × 40 mm + 2 × 80 mm × 5.5 mm + 2 × 40 mm × 5.5 mm
area = 7720 mm²
Problem
(a) Let [tex]a_1, a_2, a_3,...[/tex] be an arithmetic progression of non-zero numbers with common difference [tex]d[/tex].
(i) Show that [tex]\frac{1}{a_na_{n+1}}=\frac{1}{da_n}-\frac{1}{da_{n+1}}[/tex] for any [tex]n\geq1[/tex].
(ii) Hence show that [tex]\frac{1}{a_1a_2}+\frac{1}{a_2a_3}+\frac{1}{a_3a_4}+...+\frac{1}{a_{99}a_{100}}=\frac{99}{a_1a_{100}}[/tex].
(b) For what value of [tex]k[/tex] does [tex]\frac{1}{3\times7}+\frac{1}{7\times11}+\frac{1}{11\times15}+...+\frac{1}{k(k+4)}+\frac{2}{25}[/tex]?
(a.i) If [tex]a_1,a_2,a_3,\ldots[/tex] are in arithmetic progression, then there is a constant [tex]d[/tex] such that
[tex]a_{n+1} - a_n = d[/tex]
for all [tex]n\ge1[/tex]. In other words, the difference [tex]d[/tex] between any two consecutive terms in the sequence is always the same.
[tex]a_2-a_1 = a_3-a_2 = a_4-a_3 = \cdots = d[/tex]
Now, we can expand the target expression into partial fractions.
[tex]\dfrac1{a_na_{n+1}} = \dfrac{\alpha}{a_n} + \dfrac{\beta}{a_{n+1}}[/tex]
Combining the fractions on the right and using the recursive equation above, we have
[tex]\dfrac1{a_na_{n+1}} = \dfrac{\alpha (a_n + d) + \beta a_n}{a_n(a_n+d)} = \dfrac{(\alpha+\beta) a_n + \alpha d}{a_n a_{n+d}} \\\\ \implies \begin{cases}\alpha + \beta = 0 \\ \alpha d = 1 \end{cases} \implies \alpha = \dfrac1d, \beta = -\dfrac1d[/tex]
and hence
[tex]\dfrac1{a_n a_{n+1}} = \dfrac1{da_n} - \dfrac1{da_{n+1}}[/tex]
as required.
(a.ii) Using the previous result, the [tex]n[/tex]-th term [tex](n\ge1)[/tex] in the sum on the left is
[tex]\dfrac1{a_n a_{n+1}} = \dfrac1d \left(\dfrac1{a_n} - \dfrac1{a_{n+1}}\right)[/tex]
Expand each term in this way to reveal a telescoping sum:
[tex]\dfrac1{a_1a_2} + \dfrac1{a_2a_3} + \dfrac1{a_3a_4} + \cdots + \dfrac1{a_{99}a_{100}} \\\\ ~~~~~~~~ = \dfrac1d \left(\left(\dfrac1{a_1} - \dfrac1{a_2}\right) + \left(\dfrac1{a_2} - \dfrac1{a_3}\right) + \left(\dfrac1{a_3} - \dfrac1{a_4}\right) + \cdots + \left(\dfrac1{a_{99}} - \dfrac1{a_{100}}\right)\right) \\\\ ~~~~~~~~ = \dfrac1d \left(\dfrac1{a_1} - \dfrac1{a_{100}}\right) = \dfrac{a_{100} - a_1}{d a_1 a_{100}}[/tex]
By substitution, we can show
[tex]a_n = a_{n-1} + d = a_{n-2} + 2d = \cdots = a_1 + (n-1)d \\\\ \implies a_{100} = a_1 + 99d[/tex]
so that the last expression reduces to
[tex]\dfrac{(a_1 + 99d) - a_1}{d a_1 a_{100}} = \dfrac{99d}{d a_1 a_{100}} = \dfrac{99}{a_1 a_{100}}[/tex]
as required. More generally, it's easy to see that
[tex]\dfrac1{a_1a_2} + \dfrac1{a_2a_3} + \dfrac1{a_3a_4} + \cdots + \dfrac1{a_na_{n+1}} = \dfrac{n}{a_1a_{n+1}}[/tex]
(b) I assume you mean the equation
[tex]\dfrac1{3\times7} + \dfrac1{7\times11} + \dfrac1{11\times15} + \cdots + \dfrac1{k(k+4)} = \dfrac2{25}[/tex]
Note that the distinct factors of each denominator on the left form an arithmetic sequence.
[tex]a_1 = 3[/tex]
[tex]a_2 = 3 + 4 = 7[/tex]
[tex]a_3 = 7 + 4 = 11[/tex]
and so on, with [tex]n[/tex]-th term
[tex]a_n = 3 + (n-1)\times4 = 4n - 1[/tex]
Let [tex]a_n=k[/tex]. Using the previous general result, the left side reduces to
[tex]\dfrac1{3\times7} + \dfrac1{7\times11} + \dfrac1{11\times15} + \cdots + \dfrac1{a_na_{n+1}} = \dfrac n{3a_{n+1}} \\\\ \implies \dfrac{\frac{k+1}4}{3(k+4)} = \dfrac2{25}[/tex]
Solve for [tex]k[/tex].
[tex]\dfrac{k+1}{12k+48} = \dfrac2{25} \implies 25(k+1) = 2(12k+48) \\\\ \implies 25k + 25 = 24k + 96 \implies \boxed{k=71}[/tex]
Write an equation for the nth term of the arithmetic sequence 8, 3, -2, -7,... Find the 9th term of the sequence.
Answer:
The answer is ninth term is -32
given that £1=1.55 CHF how much cheaper is £
The £1 as a foreign exchange rate is cheaper than 1 CHF by 35%.
What is a foreign exchange rate?The foreign exchange rate refers to the unit of conversion of one currency to another.
Data and Calculations:Exchange rate = £1=1.55 CHF
Ratio of 1.55 CHF to £1 = 1 : 0.65
Cheapness of £1 = 0.35 (1 - 0.65)
Thus, £1 as a foreign exchange rate is cheaper than 1 CHF by 35%.
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Which expression is equivalent to 16^3?
Answer:
2¹²
Step-by-step explanation:
one expression equivalent to 16³ is 2¹²
think about it like this: we know that 2 to the power of 4 is 16
(2 x 2 = 4, x 2 = 8, x 2 = 16)
So for every "16" we already have 2⁴; meaning that 16³ is the same thing as 3 groups of 2⁴ -- which is 2¹²
hope this helps! have a lovely day :)
Write an equivalent expression for 3(4x+2y)+5
Answer:
12x+6y+5
Step-by-step explanation:
expand this: 3(4x+2y)=12x+6y
then add 5
12x+6y+5
Answer:
Hello! The answer is 12x + 6y + 5
Step-by-step explanation:
3(4x + 2y) + 5
Distribute:
3(4x + 2y) + 5
= 12x + 6y + 5
All like terms are combined/non-existent so the equivalent expression is: 12x + 6y + 5
Please help I want clear answers for example
AB: X = ___ Y =___
Answer:
AB: y=2x+2
CB: y=-2x+6
CD: y=1/2x-1.5
DA: y=-3x+2
Step-by-step explanation:
Answer:
AB: [tex]y=2x+2[/tex]
CB: [tex]y=-2x+6[/tex]
CD: [tex]y=\frac{1}{2}x -1.5[/tex]
DA: [tex]y=-3x+2[/tex]
Step-by-step explanation:
So when it's saying y = __ x + __ it's asking for the slope-intercept form. This is given as y=mx+b where m is the slope and b is the y-intercept. m is the slope because as x increases by 1, the y-value will increase by m, which is by definition what the slope is [tex]\frac{rise}{run}[/tex] how much the function "rises" as x "runs".
AB: See how it "rises" by 2 and only "runs by 1", thus the slope is 2/1 or 2. the last part is the y-intercept, which is the value of y when the line crosses the y-axis. If you look at the graph when it "crosses" the y-axis it's y-value is 2. So you have the equation: [tex]y=2x+2[/tex]
CB: See how it "goes down" by 4 and "runs" by 2, thus the slope is -4/2 or -2. The y-intercept isn't shown on the graph but you can calculate that by substituting known values into the slope-intercept form. So we know so far y=-2x+b since the slope was calculated, and we can take any point on the line to calculate b, for this example I'll take the point C which is (3, 0) which is (x, y) and I'll plug that in
Plug values in:
0 = -2(3) + b
0 = -6 + b
6 = b
This gives you the complete equation: [tex]y=-2x+6[/tex]
CD: see how it only "rises" by 1 but "runs" by 2, thus the slope is 1/2. The y-intercept isn't shown on the graph but you can calculate it by substituting a point into the equation For this example I'll use point C (3, 0)
0 = 3(1/2) + b
0 = 1.5 + b
-1.5 = b
This gives you the complete equation: [tex]y=\frac{1}{2}x - 1.5[/tex]
DA: see how it "decreases" by 3 but "runs" by 1, thus the slope is -3/1 or -3. The y-intercept is known and is at (0, 2) so now we plug these values in to get the equation: [tex]y=-3x+2[/tex]
What is the value of the underlined digit?
5,682, 450,003
/////^
that is the underlined digit
A. eighty thousand
B. eighty million
C. eighty-two million
D. eighty billion
The value of the underlined digit (8) in the number given is; Choice B; Eighty million.
What is the value of the underlined digit?The underlined digit in the task content is 8. On this note, it follows that the since, the number can be pronounced as; Five billion, Six hundred and Eighty-two million, Four hundred and fifty thousand, and three.
It is evident, that the place value occupied by the underlined number is; Eighty million.
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Calculus help, I am trying to do this questions.. someone teach me please 10 points
Answer:
Step-by-step explanation:
A bit late, but better than never.
24. For [tex]x\neq1[/tex], we have
[tex]\dfrac{9x^2 - x - 8}{x - 1} = \dfrac{(x-1)(9x+8)}{x-1} = 9x+8[/tex]
Then as [tex]x[/tex] approaches 1, we have
[tex]\displaystyle \lim_{x\to1} f(x) = \lim_{x\to1} (9x+8) = 17 \neq 0[/tex]
so the function is not continuous at [tex]x=1[/tex]. It is thus continuous on the interval union [tex](-\infty,0)\cup(0,\infty)[/tex]. You can also write this as "[tex]x<0[/tex] or [tex]x>0[/tex]".
25. When [tex]x[/tex] approaches 2 from the left (when [tex]x<2[/tex]), we have
[tex]\displaystyle \lim_{x\to2^-} f(x) = \lim_{x\to2} (5-x) = 3[/tex]
When [tex]x[/tex] approaches 2 from the right [tex](x>2)[/tex], we have
[tex]\displaystyle \lim_{x\to2^+} f(x) = \lim_{x\to2} (2x-3) = 1[/tex]
so the function is not continuous at [tex]x=2[/tex]. Thus it's continuous on [tex](-\infty,2)\cup(2,\infty)[/tex], or "[tex]x<2[/tex] or [tex]x>2[/tex]".
On the number line, if P points to a number 1/4 of the distance from 0.02 to 0.03, what is this number?
A 0.0021
B 0.0024
C 0.0225
D 0.025
E 0.25
Answer: C. 0.0225
Step-by-step explanation:
The distance is 0.03 - 0.02 = 0.01.
1/4 of this is 0.0025.
Adding this to 0.02, we get 0.00225.
Identifying Cross Sections of Solids
Which solids can have vertical cross sections that are circles? Check all that apply.
cones
O prisms
cylinders
O pyramids
spheres
Intro
Done
Answer:1,3,5
Step-by-step explanation:
Given a pair of complex numbers, z1 = 5 and z2 = –6i, let z3 = 3i z1 and z4 = z2 – 2. in which quadrant is z3 – z4 located?
Assuming [tex]z_1=5[/tex] and [tex]z_2=-6i[/tex], as well as [tex]z_3=3i z_1[/tex] and [tex]z_4 = z_2-2[/tex], we have
[tex]z_3 - z_4 = 3iz_1 - (z_2-2) = 3i\times5 - (-6i-2) = 15i + 6i + 2 = 2 + 21i[/tex]
Since both the real and imaginary parts are positive, [tex]z_3-z_4[/tex] belongs to the first quadrant.
Identity all obtuse angles in the drawing below
If f(x) = x2 - 1, and
g(x) = x + 2, then
f(g(x)) = [? ]x2+[?]x+[?]
Step-by-step explanation:
f.g(x) = f(x+2)
=(x+2)2 -1
=(x2+2*x*2+4)-1
=x2+4x+4-1
=x2+4x+3
In circle M. Diameters JL and HK each measure 16 centimeters.
The length of the arc in the image given is approximately: A. 3.5 cm
What is the Length of Arc?Length of arc = ∅/360 × 2πr
The missing parts of the question is shown in the image below, where we are asked to find arc length of JH.
Thus, we would have the following:
∅ = 25°
Radius (r) = 16/2 = 8 cm
Substitute the values into the formula:
Arc length JH = 25/360 × 2π(8)
Arc length JH ≈ 3.5 cm
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please help! no explanation needed
Answer: C
Step-by-step explanation:
The line is dotted, so eliminate B and D.
Also, the line has a y-intercept of 1, so this eliminates A.
This means the answer must be C.
a simple pendulum of amplitude completes 24 oscillations in one minute. find the length of the string the pendulum bob is attached
[tex]frequency = \frac{24 \: osc}{60 \: sec} = 0.4 \: osc \: per \: sec[/tex]
[tex]period = \frac{1}{frequency} = \frac{1}{0.4} = 2.5 \: sec \: per \: osc[/tex]
[tex]t = 2\pi \sqrt{ \frac{l}{g} } \\ 2.5 = 2\pi \sqrt{ \frac{l}{9.8} } \\ \frac{2.5}{2\pi} = \sqrt{ \frac{l}{9.8} } \\ ( \frac{2.5}{2\pi} ) {}^{2} = \frac{l}{9.8}[/tex]
[tex]l = ( \frac{2.5}{2\pi} ) {}^{2} \times 9.8 = 1.55148 \: meters[/tex]
Find the perimeter P of ABCD with vertices A(3,1), B(6,2), C(6,-2), and D(3,-3). Round your answer to the nearest tenth, if necessary.
Step-by-step explanation:
the distance between 2 points is per Pythagoras
c² = a² + b²
with c being the Hypotenuse (the baseline opposite of the 90° angle, which is the direct distance between the points). a and b are the legs (the x and y coordinate differences).
so,
AB² = (3-6)² + (1-2)² = 9+1 = 10
AB = sqrt(10) = 3.16227766...
BC² = (6-6)² + (2 - -2)² = 0² + 4² = 16
BC = sqrt(16) = 4
CD² = (6-3)² + (-2 - -3)² = 3² + 1² = 10
CD = sqrt(10) = 3.16227766...
DA² = (3-3)² + (-3 - 1)² = 0² + 4² = 16
DA = sqrt(16) = 4
so, the perimeter is
2×4 + 2×sqrt(10) = 8 + 6.32455532... = 14.32455532... ≈
≈ 14.3
2 4/15-z= 1 2/3
pls solve
The value of z in the equation evaluation as given in the task content is; 3/5.
What is the value of z in the equation?It follows from the task content that the value of z as in the task content can be evaluated as follows;
2 4/15-z= 1 2/3
34/15 - z = 5/3
Hence, z = 34/15 - 5/3
z = (34-25)/15 = 9/15
z = 3/5
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Which of the following equations have complex roots?
Answer:
D
Step-by-step explanation:
Subtracting 2 from both sides and then dividing both sides by 3 gives that x² = -2/3, which indicates the roots are complex.
The quadrilaterals ABCD and JKLM are similar.
Find the length x of LM.
D
5
4
A
3
B
2
C
M
6
X
J
3.6
M
2.4
7
Answer: 4.8
Step-by-step explanation:
Corresponding sides of similar figures are proportional, so
[tex]\frac{x}[4}=\frac{6}[5}\\\\x=4\left(\frac{6}{5} \right)=\boxed{4.8}[/tex]
Lydia has half of her investments in stock paying an 11% dividend and the other half in a stock paying 14% interest. If her total annual interest is $440, how much does she have invested?
Answer
The amount that Lydia has earned on her investment is $198 on the stock paying the 11% dividend and $252 on the stock paying 14% interest for a total earned of $450 on her $3600 investment.
Calculations and ParametersLet x represent half of her total investment.
The total annual interest earned is $450.
This is represented by the following equation:
0.11x + 0.14x= 450
Collect the like terms together
0.25x = 450
x= 1800
Therefore, half of her total investment is $1800 so her total investment is $3600.
Check the solution:
0.11∗1800 + 0.14 ∗ 1800 = 450
198 + 252 = 450
Therefore, she earned $198 on the stock paying the 11% dividend, and $252 on the stock paying 14% interest for a total earned of $450 on her $3600 investment.
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Question 8 of 25
Which of the following is the quotient of the rational expressions shown here?
Answer:
B. (5x+5)/(x²-2x)
Step-by-step explanation:
As with numerical fractions, division by a rational expression is equivalent to multiplication by its reciprocal.
As a multiplication problem[tex]\dfrac{5}{x-2}\div\dfrac{x}{x+1}=\dfrac{5}{x-2}\times\dfrac{x+1}{x}[/tex]
Product of rational expressionsAs with numerical fractions, the product is the product of numerators, divided by the product of denominators.
[tex]=\dfrac{5(x+1)}{x(x-2)}=\boxed{\dfrac{5x+5}{x^2-2x}}[/tex]
Based on the data in the two-way table, the probability of being 25-35 years and having a hemoglobin level above 11 is __ a.29 b.34 c.54
The probability of having a hemoglobin level above 11 is __ a.25 b.3 c.36
Being 25-35 years and having a hemoglobin level above 11 __ dependent on each other. a.are b.are not
The probability of being 25-35 years and having a haemoglobin level above 11 is 34%.
The probability of having a haemoglobin level above 11 is 36%.
Being 25-35 years and having a hemoglobin level above 11 are not dependent on each other.
What are the probabilities?Probability determines the odds that a random event would occur. The odds of the event happening lie between 0 and 1.
The probability of being 25-35 years and having a haemoglobin level above 11 = number of people between 25 - 35 that have a level above 11 / total number of people between 25 - 35
44 / 128 = 34%
The probability of having a haemoglobin level above 11 = number of people with a level above 11 / total number of respondents
153 / 429 = 36%
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Answer two questions about Systems A and B:
A
4x+16y=12
x+2y=−9
B
4x+16y=12
x+4y=3
1) How can we get System B from System A?
Choose 1 answer:
(Choice A, Checked, Correct)
CORRECT (SELECTED)
Replace one equation with a multiple of the other equation
For the given systems the answers are:
1) "Replace one equation with a multiple of the other equation".
2) The systems are not equivalent.
How can we get System B from System A?Here we have the systems of equations:
A:
4x + 16y = 12
x +2y = -9
B:
4x + 16y = 12
x + 4y = 3
Notice that the first equation is the same in both systems, but the second is different.
In A we have:
x + 2y = -9
In B we have:
x + 4y = 3
So we need to add 2y on the left side of A, and 12 on the right side of A to get the equation in B.
Or we can just take the other equation in A, divide it by 4, and replace it.
So we need to replace the second equation in A for other equation, then the correct option is:
"Replace one equation with a multiple of the other equation".
Because we are replacing an equation in A by other to get system B, we conclude that systems are not equivalent.
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Answer:
khan
Step-by-step explanation:
is 66 a perfect cube with prime factorization
Answer:
no
Step-by-step explanation:
Prime factorization of 66
[tex]66=[/tex][tex]2[/tex] × [tex]3[/tex] × [tex]11[/tex]
the perfect cubes are the numbers that possess exact cubic roots
[tex]\sqrt[3]{66}[/tex] ≈ [tex]4.04[/tex]
66 is not a perfect cube
Hope this helps
A bag of mixed nuts contains cashews, pistachios, and almonds. There are 1,000 total nuts in the bag, and there are 100 less almonds than pistachios. The cashews weigh 3 g, pistachios weigh 4 g, and almonds weigh 5 g. If the bag weighs 3.7 kg, fi d out how many of each type of nut is in the bag.
Answer:
Below in bold.
Step-by-step explanation:
Let the number of pistachios be x then the number of almonds = x-100.
The number of cashews = 1000 - x - (x - 100) = 1100-2x.
From the info given
3(1100-2x) + 4x + 5(x - 100) = 3700
3300 - 6x + 4x + 5x - 500 = 3700
3x = 3700 - 3300 + 500
3x = 900
x = 300
So there are 300 pistachios, 200 almonds and 500 cashews in the bag.
There are 300 pistachios, 200 almonds, and 500 cashews in the bag of mixed nuts.
To find the number of each type of nut in the bag, we can set up a system of equations based on the given information.
Let's assume there are x pistachios in the bag. According to the problem, there are 100 fewer almonds than pistachios, so the number of almonds will be (x - 100).
Now, the total number of nuts is 1,000:
x (pistachios) + (x - 100) (almonds) + cashews = 1000
The total weight of the nuts in the bag can be calculated as follows:
3g (cashews) * number of cashews + 4g (pistachios) * x + 5g (almonds) * (x - 100) = Total weight in grams.
Since we know that the bag weighs 3.7 kg, we need to convert the total weight to grams and set it equal to 3.7 kg * 1000g/kg = 3700g.
Now we have two equations:
x + (x - 100) + cashews = 1000
3g * cashews + 4g * x + 5g * (x - 100) = 3700
Let's simplify equation 1:
2x - 100 + cashews = 1000
2x + cashews = 1100
cashews = 1100 - 2x
Now, let's substitute the expression for cashews in equation 2:
3g * (1100 - 2x) + 4g * x + 5g * (x - 100) = 3700
Now, distribute the weights:
3300g - 6g * x + 4g * x + 5g * x - 500g = 3700
Combine like terms:
-6g * x + 4g * x + 5g * x = 3700 - 3300 + 500
3g * x = 900
x = 900 / 3
x = 300
Now that we have the value of x (number of pistachios), let's find the number of almonds:
almonds = x - 100
almonds = 300 - 100
almonds = 200
Now, let's find the number of cashews using the total nut count:
cashews = 1000 - (pistachios + almonds)
cashews = 1000 - (300 + 200)
cashews = 1000 - 500
cashews = 500
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An insurance policy sells for $600. Based on past data, an average of 1 in 50 policy holders
will file a $5,000 claim, an average of 1 in 100 policyholders will file a $10,000 claim, and an average
of 1 in 200 policyholders will file a $30,000 claim.
(a) Find the expected value (to the company) per policy sold.
(b) If the company sells 10,000 policies, what is the expected profit or loss
Using a discrete distribution, we have that:
a) The expected value is of $250 per policy sold.
b) The expected profit for 10,000 policies is of $2,500,000.
What is the mean of a discrete distribution?The expected value of a discrete distribution is given by the sum of each outcome multiplied by it's respective probability.
Considering that an insurance profit sells for $600, the distribution of the company's earnings is given as follows:
P(X = -4400) = 0.02.P(X = -9400) = 0.01.P(X = -29400) = 0.005.P(X = 600) = 0.965.Hence the expected value for a policy sold is given by:
E(X) = 600(0.965) - 4400(0.02) -9400(0.01) - 29400(0.005) = 250.
For 10,000 policies, the expected profit is given by:
E = 10000 x 250 = 2,500,000.
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Pls help!! Geometry
Find the perimeter of the figure.
parallelogram has 2 pairs of congruent sides and the opposite angles are equal.
Answer:
True.
Step-by-step explanation:
In order for a quadrilateral to be a parallelogram, the top and bottom sides must be congruent, and the left and right must be congruent.
Opposite angles on either side are equal.
A roller coaster’s height is given by the equation h = –.025t2 + 4t + 50, where t represents the time in seconds. How long will it take riders to pass over the hill and reach ground level? Hint: Set h = 0.
It will take the roller coaster 171.652 seconds to pass over the hill and reach ground level
How to determine the time to hit the ground?The function is given as:
h = –.025t^2 + 4t + 50
Set h = 0.
So, we have:
–.025t^2 + 4t + 50 = 0
Using a graphing calculator, we have:
t = -11.652 and t = 171.652
Time cannot be negative.
So, we have:
t = 171.652
Hence, it will take the roller coaster 171.652 seconds to pass over the hill and reach ground level
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