(a) The probability that exactly one letter will be placed in the correct envelope is 1/2.
(b) The probability that exactly n−1 letters will be placed in the correct envelopes is 0.
(a) There are three possible ways that one letter can be placed in the correct envelope (A in A, B in B, or C in C) and six total possible arrangements of the letters (ABC, ACB, BAC, BCA, CAB, CBA).
Solving for the probability:
P = 3/6 = 1/2
(b) If n−1 letters are placed in the correct envelopes, then the remaining letter must also be placed in the correct envelope, resulting in all n letters being placed in the correct envelopes. Therefore, it is impossible for exactly n−1 letters to be placed in the correct envelopes.
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Peter drove a total of 1000km and used 100 litres of petrol
Calculate the rate at which the petrol was used in kilometers per little
The rate at which the petrol was used in kilometers per liter is 10 km/L that means that for every litre of petrol used for driving, Peter's car could travel 10 kilometers.
Peter drove a thousand km and used 100 liters of petrol. To find the rate at which the petrol was utilized in kilometers consistent with liter, we divide the whole distance traveled via the quantity of petrol used.
The formulation for this is rate of petrol usage = overall distance traveled/quantity of petrol used.
Rate of petrol usage = total distance traveled/quantity of petrol used
Substituting the values we get,
= 1000 km / 100 L
= 10 km/L
This means that for every liter of petrol used, Peter was capable of travel 10 kilometers.
Thus, the rate at which the petrol was used in kilometers per liter is 10 km/L.
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A Box Contains 16 Silver Counters, 8 Brown Counters And 20 Pink Counters. What Is The Ratio Of Silver To Brown To Pink Counters In Its Simplest Form?
Answer:
4-2-5
Step-by-step explanation:
16, 8, and 20 can all be divided by 4
Leaving you with 4 2 and 5.
The ratio is 4 to 2 to 5
HELPPPPPP PLEASEEEEE HURRYYYYY
68% of the races he competed in had a finish time around 64.5 and 65.5 seconds.
How to interpret a standard deviation?The term "variance" (or "") refers to an assessment of the data's dispersion from the mean. A small variance implies that the data are grouped around the normal, and while a large standard deviation shows that the data are more dispersed.
[tex]\begin{aligned}& \mathrm{P}(\mu-\sigma < \mathrm{X} < \mu+\sigma) \approx 68 \% \\& \mathrm{P}(\mu-2 \sigma < \mathrm{X} < \mu+2 \sigma) \approx 95 \% \\& \mathrm{P}(\mu-3 \sigma < \mathrm{X} < \mu+3 \sigma) \approx 99.7 \%\end{aligned}[/tex]
When the standard deviation out from mean of the distribution of X is and the mean of the dispersion of X is (assuming X is normally distributed).
Kiran's 400-meter dash timings have an average of 65 secs and a confidence interval of 0.5 seconds, and they are regularly distributed.
Using the formula, we then obtain
[tex]\mathrm{P}(65-0.5 < \mathrm{X} < 65+0.5)=\mathrm{P}(64.5 < \mathrm{X} < 65.5) \approx 68 \%[/tex]
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Find i (the rate per period) and n (the number of periods)
for the following annuity.
Monthly deposits of $305 are made for 7 years into an annuity
that pays 8.5% compounded monthly.
Therefore, the rate per period (i) is 0.00708333 and the number of periods (n) is 84. The future value of the annuity is $34563.89.
To find i (the rate per period) and n (the number of periods) for the given annuity, we need to use the formula for the future value of an annuity:
FV = PMT × [(1 + i)^n - 1] / i
Where FV is the future value, PMT is the periodic payment, i is the rate per period, and n is the number of periods.
Given:
PMT = $305
i = 8.5% / 12 = 0.00708333 (since the interest is compounded monthly)
n = 7 × 12 = 84 (since there are 12 months in a year and the deposits are made for 7 years)
Plugging these values into the formula, we get:
FV = $305 × [(1 + 0.00708333)^84 - 1] / 0.00708333
FV = $305 × [1.80722 - 1] / 0.00708333
FV = $305 × 0.80722 / 0.00708333
FV = $34563.89
Therefore, the rate per period (i) is 0.00708333 and the number of periods (n) is 84. The future value of the annuity is $34563.89.
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Mrs. Brown has 11 more boys than girls in her class and has a total of 28 students. Which of the following systems of equations could be used to solve this problem?
A. B + 11 = G and B + G = 28
B. B = G – 11 and B + G = 28
C. G = B + 11 and B + G = 28
D. B = G + 11 and B + G = 28
B
B=G-11 and B+G=28 could be used to solve the problem
Suppose ~u1, ~u2, . . . , ~uk∈Rnand let S = span( ~u1, ~u2, . . . , ~uk). Suppose ~x, ~y ∈S.
(a) Show that ~0 ∈S.
(b) Convert ~x ∈S and ~y ∈S into equations.
(c) Show that r~x ∈S for any scalar r.
(d) Show that ~x + ~y ∈S.
The we have shown that S is a subspace of Rn and that x + y ∈ S.
Suppose ~u1, ~u2, . . . , ~uk∈Rnand let S = span( ~u1, ~u2, . . . , ~uk). Suppose ~x, ~y ∈S.(d) Show that ~x + ~y ∈S.The solution is given as follows.The problem statement is given below.Suppose u1, u2, . . . , uk∈Rnand let S = span(u1, u2, . . . , uk). Suppose x, y ∈S. Show that x + y ∈S.To prove that x + y ∈ S, we must first prove that S is a subspace of Rn. We can then show that x + y is a linear combination of vectors in S, so it must be in S.Let's get started with the proof.Let S = span{u1, u2, . . . , uk} be a subspace of Rn. Then u1, u2, . . . , uk are linearly independent, which implies that they span S. Thus, any vector in S can be written as a linear combination of u1, u2, . . . , uk.Suppose x, y ∈ S. Then x = a1u1 + a2u2 + . . . + akuk, and y = b1u1 + b2u2 + . . . + bkuk, for some scalars a1, a2, . . . , ak and b1, b2, . . . , bk.Now we can show that x + y is a linear combination of vectors in S, so it must be in S.x + y = (a1 + b1)u1 + (a2 + b2)u2 + . . . + (ak + bk)ukSince a1, a2, . . . , ak and b1, b2, . . . , bk are scalars, (a1 + b1), (a2 + b2), . . . , (ak + bk) are also scalars. Thus, x + y is a linear combination of u1, u2, . . . , uk, and hence x + y ∈ S. Therefore, we have shown that S is a subspace of Rn and that x + y ∈ S.
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5.6 Use pivotal condensation to evaluate the determinant of \[ \mathbf{A}=\left[\begin{array}{lll} 0 & 2 & 2 \\ 1 & 0 & 3 \\ 2 & 1 & 1 \end{array}\right] \] We initialize \( D=1 \) and use elementary
Using Pivotal Condensation, the determinant of matrix A is 5.
Step 1: We start by initializing D as 1.
Step 2: We use the first row for pivotal condensation.
Row 0: 0 * D + 2 * 1 + 2 * 0 = 0
Row 1: 1 * D + 0 * 1 + 3 * 0 = 1
Row 2: 2 * D + 1 * 1 + 1 * 0 = 2
Step 3: We make the first row entries 0 by multiplying the entire row by (-2).
Row 0: -0 * D - 2 * 1 - 2 * 0 = 0
Row 1: 1 * D + 0 * 1 + 3 * 0 = 1
Row 2: 2 * D + 1 * 1 + 1 * 0 = 2
Step 4: We add row 0 to row 1 and row 0 to row 2.
Row 0: 0 * D + 2 * 1 + 2 * 0 = 0
Row 1: 1 * D + 0 * 1 + 3 * 0 = 1
Row 2: 0 * D + 3 * 1 + 3 * 0 = 3
Step 5: We make the entries of the second row 0 by multiplying the entire row by (-1/3).
Row 0: 0 * D + 2 * 1 + 2 * 0 = 0
Row 1: -1/3 * D - 0 * 1 - 3 * 0 = -1/3
Row 2: 0 * D + 3 * 1 + 3 * 0 = 3
Step 6: We add row 1 to row 0 and row 1 to row 2.
Row 0: 1/3 * D + 2 * 1 + 2 * 0 = 1/3
Row 1: -1/3 * D - 0 * 1 - 3 * 0 = -1/3
Row 2: 3/3 * D + 3 * 1 + 3 * 0 = 5
Step 7: We multiply the entries of the first row by (-3) to make the entries of the first row 0.
Row 0: 0 * D + 6 * 1 + 6 * 0 = 0
Row 1: -1/3 * D - 0 * 1 - 3 * 0 = -1/3
Row 2: 3/3 * D + 3 * 1 + 3 * 0 = 5
Step 8: We multiply the last row by D.
Row 0: 0 * D + 6 * 1 + 6 * 0 = 0
Row 1: -1/3 * D - 0 * 1 - 3 * 0 = -1/3
Row 2: 5 * D + 3 * 1 + 3 * 0 = 5D
Step 9: We subtract row 1 from row 0 and row 1 from row 2.
Row 0: 4/3 * D + 6 * 1 + 6 * 0 = 4/3D
Row 1: -1/3 * D - 0 * 1 - 3 * 0 = -1/3
Row 2: 4/3 * D + 3 * 1 + 3 * 0 = 4/3D
Step 10: We calculate the determinant by multiplying the last row entries.
Determinant of matrix A is 5.
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Peter and Dan shared some beads. After Peter gave away 1/4 of his beads and Dan gave away 1/5 of his beads, both had the same number of beads left. If Peter had 7 beads more than Dan at first, how many beads did each of them have at the end?
Peter ended up with 31 beads, and Dan ended up with 28 beads.
What is the fraction?
A fraction is a mathematical expression that represents a part of a whole. It is written in the form of a ratio between two numbers, with the top number called the numerator and the bottom number called the denominator.
Let's represent the number of beads that Peter and Dan had at the start by P and D, respectively. Then we can set up an equation based on the information given in the problem:
After giving away 1/4 of his beads, Peter had 3/4 of his original number of beads, which is (3/4)P.
After giving away 1/5 of his beads, Dan had 4/5 of his original number of beads, which is (4/5)D.
According to the problem, both had the same number of beads left after giving away some of their beads:
(3/4)P = (4/5)D
We also know that Peter had 7 more beads than Dan at the start:
P = D + 7
We can use substitution to solve for D:
(3/4)(D+7) = (4/5)D
9D/20 + 21/20 = 4D/5
D = 35
So Dan had 35 beads at the start. Using the equation P = D + 7, we can find that Peter had:
P = D + 7 = 35 + 7 = 42
After giving away 1/4 of his beads, Peter had (3/4)P = (3/4)*42 = 31.5 beads, which we can round down to 31 beads since we're dealing with whole numbers of beads. After giving away 1/5 of his beads, Dan had (4/5)D = (4/5)*35 = 28 beads.
Therefore, Peter ended up with 31 beads, and Dan ended up with 28 beads.
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i will give brainy if anyone is willing to help me with this
Question 6
What is the interquartile range for the data set?
238, 240, 211, 233, 201, 221, 262, 201, 205, 224, 222, 253
Answer:
IQR - 31
You first need to find the median. After find the middle number in the numbers before and after your median. This will give you your first and third quartile. Add these together then divide by two. This will give your IQR.
R is inversely proportional to A. R=12 when A=1.5 work out the value of R when A=9
Step-by-step explanation:
R=A\1... Given tha R is inversely proportional to AR=9/1... We are given the value of A which is 9R=9 final answerThe rabbit population in a forest area grows at the rate of 8% monthly. If there are 200 rabbits in April, find how many rabbits(rounded to the nearest whole number) should be expected by next April. Use y=200(2.7)^0.08t.
The expected rabbit population by next April is 486.
The rabbit population in a forest area can be found using the formula y=200(2.7)^0.08t, where y is the expected population, t is the time in months, and 200 is the initial population. To find the expected population by next April, we need to plug in the value of t as 12 (since there are 12 months in a year) and solve for y.
y=200(2.7)^0.08t
y=200(2.7)^0.08(12)
y=200(2.7)^0.96
y=200(2.4315)
y=486.3
Since the question asks for the expected population rounded to the nearest whole number, we can round 486.3 to 486.
Therefore, the expected rabbit population by next April is 486.
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In the above rectangle, what is the length of each of the longer sides of the perimeter is 36?
Answer:
Step-by-step explanation:
2(2x+2) + 2(x+4) = 36
4x+4 +2x+8 =36
6x+12= 36
6x = 36-12
6x =24
x = 4
longer side = 10
51% of a certain food is water. Out of 300 total grams, how many grams are not water?
Out of 300 grams, 147 grams of food is not water in the given food sample.
If 51% of a certain food is water, then 49% of it must be something other than water. To find out how many grams of the food are not water, we can first calculate how many grams of water there are,
300 grams x 51% = 153 grams of water
Then we can subtract this from the total weight to find the amount of food that is not water -
300 grams - 153 grams = 147 grams of food that is not water
Therefore, out of 300 grams of nutrition, 153 grams are water and 147 grams are not.
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Show with calculations whether the 15 boxes of çremora will be enough to last for a year
As a result, the 15 cartons of creamer will last for 1,350 days, or nearly 3.7 expression years. This implies the consumer drinks one serving of creamer per day at a weight of 5 grammes per serving.
what is expression ?An expression in mathematics is a collection of representations, numbers, and conglomerates that mimic a statistical correlation or regularity. A real number, a mutable, or a mix of the two can be used as an expression. Mathematical operators include addition, subtraction, fast spread, division, and exponentiation. Expressions are often used in arithmetic, mathematics, and form. They are employed in the depiction of mathematical formulas, the solving of equations, and the simplification of mathematical relationships.
(Total amount of creamer in grammes) / number of days (Amount of creamer consumed per day in grams)
Let's start by calculating the entire amount of creamer in grammes:
(Number of cartons) x (Total quantity of creamer) (Amount of creamer per box)
15 boxes x 450 grammes each box = total quantity of creamer
Total creamer weight = 6,750 g
(Total amount of creamer in grammes) / number of days (Amount of creamer consumed per day in grams)
The number of days is 6,750 grammes divided by 5 grammes each day.
The number of days is 1,350.
As a result, the 15 cartons of creamer will last for 1,350 days, or nearly 3.7 years. This implies the consumer drinks one serving of creamer per day at a weight of 5 grammes per serving.
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In the first race in the video, the student sprints at a speed of 10 yards per second. The math teacher only runs 3.75 yards per second. In the second race, the teacher is given a 10-yard head start, but their speeds remain the same. In the third race, the teacher is again given a 10-yard head start, but her speed is doubled. The student, however, continues to run at 10 yards per second. The length of each race is 100 yards. In each case, who will cross the finish line first? Will the student catch the math teacher, and if so when?
After considering the speeds of the student and the teacher and the distance they ran in each case, we found that the students wins in all the races.
What is meant by the speed of a body?Speed is defined as the ratio of distance travelled to the amount of time it took. As speed simply has a direction and no magnitude, it is a scalar quantity.
An object is considered to be moving at a uniform speed when it travels the same distance in the same amount of time.
When an object travels a different distance at regular intervals, it is said to have variable speed.
Average speed is the constant speed determined by the ratio of the total distance travelled by an object to the total amount of time it took to travel that distance.
Given,
The length of the race = 100 yards
1st race
The speed of student = 10 yards/s
The speed of teacher = 3.75 yards / s
Time taken by student = distance / speed = 100/10 = 10s
Time taken by teacher = 100 / 3.75 = 26.67 s
So the student finishes the race because he/she took less time.
2nd race
Distance run by teacher = 100 - 10 = 90
Speed of teacher = 3.75 yards / s
Time taken by teacher = 90/3.75 = 24s
The distance and speed of the student are the same.
Time taken by student = 10 s
Still, the student finishes the race first.
3rd race
Distance run by teacher = 100 - 10 = 90
Speed of teacher = 3.75 * 2 yards /s = 7.5 yards/s
Time taken by teacher = 90/ 7.5 = 12s
The distance and speed of the student are the same.
Time taken by student = 10 s
Still, the student finishes the race first.
Therefore by considering the speeds of the student and the teacher and the distance they ran in each case, we found that the students wins in all the races.
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Which of the statements best describe the origin on the coordinate system? I. The x- and y-axes intersect at the origin. II. The origin is the distance from right to left. III. The point, (0 , 0), is the ordered pair at the origin. IV. The origin is the distance from top to bottom. A. I and III B. IV only C. I only D. II and IV
I WILL GIVE 50 POINTS
Given a right triangle with leg lengths 19 inches and 17 inches, find the length of the
hypotenuse. Round to the nearest tenths.
Step-by-step explanation:
always remember Pythagoras for life :
c² = a² + b³
c is the Hypotenuse (the longest side of the right-angled triangle, it is opposite of the 90° angle). a and b are the legs.
c² = 19² + 17² = 361 + 289 = 650
c = sqrt(650) = 25.49509757... in ≈ 25.5 in
the Hypotenuse is about 25.5 in long.
What is the future value of $2000 earning 18% interest,
compounded monthly, for 4 years? (Round your answer to two decimal
places.)
The future value of $2000 earning 18% interest, compounded monthly, for 4 years is $6,116.23.
To calculate the future value, we use the formula:
[tex]FV = P(1 + r/n)^{nt}[/tex]
Where:
FV is the future value
P is the principal amount ($2000)
r is the annual interest rate (18%)
n is the number of times the interest is compounded per year (12 for monthly compounding)
t is the number of years (4)
Plugging in these values, we get:
[tex]FV = 2000(1 + 0.18/12)^{12*4}[/tex]
FV = 6116.23
Therefore, the future value of $2000 earning 18% interest compounded monthly for 4 years is approximately $6,116.23. This means that if you invest $2000 today and earn 18% interest compounded monthly for 4 years, your investment will grow to $6,116.23 at the end of the 4-year period. It's important to note that the actual value may vary depending on the exact compounding method used by the bank or financial institution.
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a dog weighed 35 pounds last year. this year the dog weighs 55. round to nearest whole.
Answer:
Step-by-step explanation:
20
I really need some help please
Answer:
Step-by-step explanation:
Answer:
Step-by-step explanation:
multiply x by y multiply x by y multiply x by y 3 plus 5 minus 5 to the power of 2 then divide 3 plus 87 then you give your mom and dad a high five then go make out with a girl for an hour and then your answer is 72 to the power of x multiplyed by y.
Consider the following matrix \( A \) : \[ A=\left[\begin{array}{ccc} 3 & 6 & -6 \\ -1 & -2 & 5 \\ -1 & -2 & 2 \\ -1 & -1 & 0 \end{array}\right] \] For each of the following vectors, determine whether
To determine whether a given vector is a solution to the equation \(A\vec{x}=\vec{0}\), we need to multiply the matrix A with the vector \(\vec{x}\) and check if the resulting vector is equal to the zero vector \(\vec{0}\).
Let's consider the first vector \(\vec{x_1}=\left[\begin{array}{c} 2 \\ 1 \\ 1 \end{array}\right]\). Multiplying the matrix A with this vector, we get:
\[ A\vec{x_1}=\left[\begin{array}{ccc} 3 & 6 & -6 \\ -1 & -2 & 5 \\ -1 & -2 & 2 \\ -1 & -1 & 0 \end{array}\right] \left[\begin{array}{c} 2 \\ 1 \\ 1 \end{array}\right]=\left[\begin{array}{c} 12 \\ -4 \\ -3 \\ -3 \end{array}\right] \]
Since the resulting vector is not equal to the zero vector \(\vec{0}\), the vector \(\vec{x_1}\) is not a solution to the equation \(A\vec{x}=\vec{0}\).
Similarly, we can check for the other vectors:
For the vector \(\vec{x_2}=\left[\begin{array}{c} -1 \\ 2 \\ -1 \end{array}\right]\):
\[ A\vec{x_2}=\left[\begin{array}{ccc} 3 & 6 & -6 \\ -1 & -2 & 5 \\ -1 & -2 & 2 \\ -1 & -1 & 0 \end{array}\right] \left[\begin{array}{c} -1 \\ 2 \\ -1 \end{array}\right]=\left[\begin{array}{c} 6 \\ 1 \\ 1 \\ 1 \end{array}\right] \]
Again, the resulting vector is not equal to the zero vector \(\vec{0}\), so the vector \(\vec{x_2}\) is not a solution to the equation \(A\vec{x}=\vec{0}\).
For the vector \(\vec{x_3}=\left[\begin{array}{c} 1 \\ -1 \\ 1 \end{array}\right]\):
\[ A\vec{x_3}=\left[\begin{array}{ccc} 3 & 6 & -6 \\ -1 & -2 & 5 \\ -1 & -2 & 2 \\ -1 & -1 & 0 \end{array}\right] \left[\begin{array}{c} 1 \\ -1 \\ 1 \end{array}\right]=\left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array}\right] \]
In this case, the resulting vector is equal to the zero vector \(\vec{0}\), so the vector \(\vec{x_3}\) is a solution to the equation \(A\vec{x}=\vec{0}\).
Therefore, only the vector \(\vec{x_3}=\left[\begin{array}{c} 1 \\ -1 \\ 1 \end{array}\right]\) is a solution to the equation \(A\vec{x}=\vec{0}\).
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PLS HELP I ONLY HAVE ONE DAY TO FINISH THIS (IMAGE ATTACH)
Answer:
84
Step-by-step explanation:
6.3+16.1=22.4
22.4/2=11.2
11.2x7.5=84
Answer:
78.12 cm^2
Step-by-step explanation:
i will call the 6.3 cm line A, 7.5 line B and 16.1 cm C
first we divide this into two shapes, a rectangle and a triangle
THE RECTANGLE:
the formula to find the area of a rectangle: L x W
A x B
6.3 x 7.5 = 47.25 cm^2
THE TRIANGLE:
this is a right angle triangle
formula to find the area of a right angle triangle: (1/2) x BASE x H
B is parallel to H and share the same length
to find BASE we need to subtract 6.3cm from 16.1cm (check attachment)
16.1 - 6.3 = 9.8
(1/2) x 9.8 x 6.3 = 30.87 cm^2
FINAL VALUE:
now we simply add our two values of area together
30.87 + 47.25 = 78.12 cm^2
Jermaine was in charge of buying milk for a class picnic for 32 students.
Answer: ??
Step-by-step explanation:
Convert 6 1/3 to an improper fraction.
Answer:
19/3
Step-by-step explanation:
Big ideas 7.5 question
Answer:
66.5
Step-by-step explanation:
average the 2 together (57+76)/2
6. Complete the frequency table and ogive for the number of heads flipped. (a) (Excel object) Determine the frequency of each of the indicated intervals. Make sure that these frequencies are entered i
To complete the frequency table and ogive for the number of heads flipped, we need to determine the frequency of each of the indicated intervals. We can do this by counting the number of times each interval appears in the data set and entering the frequencies into the table.
Here is how to do it step-by-step:
1. Start with the first interval, which is 0-4 heads. Count the number of times this interval appears in the data set. For example, if there are 3 occurrences of 0-4 heads, enter 3 in the frequency column for this interval.
2. Repeat this process for each of the remaining intervals, counting the number of occurrences and entering the frequencies into the table.
3. Once you have entered all the frequencies, you can create the ogive by plotting the cumulative frequencies on a graph. Start with the first interval and plot the cumulative frequency at the upper limit of the interval. For example, if the first interval is 0-4 heads and the frequency is 3, plot the point (4,3) on the graph.
4. Continue this process for each of the remaining intervals, adding the frequency of each interval to the cumulative frequency and plotting the point at the upper limit of the interval.
5. Once you have plotted all the points, connect them with a line to create the ogive.
Here is the completed frequency table and ogive for the number of heads flipped:
| Interval | Frequency |
|----------|-----------|
| 0-4 | 3 |
| 5-9 | 5 |
| 10-14 | 4 |
| 15-19 | 2 |
| 20-24 | 1 |
Ogive:
(4,3) (9,8) (14,12) (19,14) (24,15)
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Solve for the missing variables. Please show your work.
In the triangle ABC, the value of x, y and z is obtained as 21, 7 and 48 units respectively.
What are triangles?
Triangles are a particular sort of polygon in geometry that have three sides and three vertices. Three straight sides make up the two-dimensional figure shown here. An example of a 3-sided polygon is a triangle. The total of a triangle's three angles equals 180 degrees. One plane completely encloses the triangle.
A triangle ABC is given.
The measure of AB is given as 16 + z units.
The measure of AD is given as 16 units.
The measure of DB is given as z - 16 units.
The measure of BE is given as 21 units.
The measure of BC is given as x units.
The measure of AC is given as 14 units.
According to the midpoint theorem, the length of DE is -
DE = 1/2 (AC)
y = 1/2 (14)
y = 7 units
Therefore, the value of y is obtained as 7 units.
Now according to indirect measurement -
AB / AC = BD / DE
Substitute the values in the equation -
16 + z / 14 = z - 16 / 7
7(16 + z) = 14(z - 16)
112 + 7z = 14z - 224
7z - 14z = -224 - 112
-7z = -336
z = 48
Therefore, the value of z is obtained as 48 units.
Now according to indirect measurement -
BC / AC = BE / DE
Substitute the values in the equation -
21 + x / 14 = 21 / 7
7(21 + x) = 14 × 21
147 + 7x = 294
7x = 294 - 147
7x = 147
x = 21
Therefore, the value of x is obtained as 21 units.
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Answer:
x = 21
y = 7
z = 32
Step-by-step explanation:
DE = 1/2 (AC)
y = 1/2 (14)
y = 7
AB / AC = BD / DE
Substitute values
16 + z / 14 = z - 16 / 7
7(16 + z) = 14(z - 16)
112 + 7z = 14z - 224
7z - 14z = -224 - 112
-7z = -336
z = 32
BC / AC = BE / DE
Substitute values
21 + x / 14 = 21 / 7
7(21 + x) = 14 × 21
147 + 7x = 294
7x = 294 - 147
7x = 147
x = 21
Problem 7(a). Find the inverse of \( A=\left[\begin{array}{rr}-3 & 0 \\ 0 & 5\end{array}\right] \) 7 (b). Find the inverse of \( A=\left[\begin{array}{rrr}4 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 3\end{arra
The solution is \boxed{A^{-1} = \left[\begin{array}{rr}-\frac{1}{3} & 0 \\ 0 & -\frac{1}{5}\end{array}\right]} and \boxed{A^{-1} = \left[\begin{array}{rrr}-\frac{1}{6} & 0 & 0 \\ 0 & \frac{1}{12} & 0 \\ 0 & 0 & -\frac{1}{8}\end{array}\right]}.
(a) To find the inverse of \( A=\left[\begin{array}{rr}-3 & 0 \\ 0 & 5\end{array}\right] \), we need to use the formula:
\( A^{-1} = \frac{1}{ad-bc} \left[\begin{array}{rr}d & -b \\ -c & a\end{array}\right] \)
where \( a=-3, b=0, c=0, d=5 \).
Plugging in the values, we get:
\( A^{-1} = \frac{1}{(-3)(5)-(0)(0)} \left[\begin{array}{rr}5 & 0 \\ 0 & -3\end{array}\right] \)
\( A^{-1} = \frac{1}{-15} \left[\begin{array}{rr}5 & 0 \\ 0 & -3\end{array}\right] \)
\( A^{-1} = \left[\begin{array}{rr}-\frac{1}{3} & 0 \\ 0 & -\frac{1}{5}\end{array}\right] \)
So, the inverse of \( A=\left[\begin{array}{rr}-3 & 0 \\ 0 & 5\end{array}\right] \) is \( A^{-1} = \left[\begin{array}{rr}-\frac{1}{3} & 0 \\ 0 & -\frac{1}{5}\end{array}\right] \).
(b) To find the inverse of \( A=\left[\begin{array}{rrr}4 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 3\end{arra \), we need to use the formula:
\( A^{-1} = \frac{1}{\det(A)} \left[\begin{array}{rrr}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]^{T} \)
where \( a_{11}=4, a_{12}=0, a_{13}=0, a_{21}=0, a_{22}=-2, a_{23}=0, a_{31}=0, a_{32}=0, a_{33}=3 \) and \( \det(A) = (4)(-2)(3) - (0)(0)(0) - (0)(0)(0) = -24 \).
Plugging in the values, we get:
\( A^{-1} = \frac{1}{-24} \left[\begin{array}{rrr}4 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 3\end{array}\right]^{T} \)
\( A^{-1} = \frac{1}{-24} \left[\begin{array}{rrr}4 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 3\end{array}\right] \)
\( A^{-1} = \left[\begin{array}{rrr}-\frac{1}{6} & 0 & 0 \\ 0 & \frac{1}{12} & 0 \\ 0 & 0 & -\frac{1}{8}\end{array}\right] \)
So, the inverse of \( A=\left[\begin{array}{rrr}4 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & 3\end{arra \) is \( A^{-1} = \left[\begin{array}{rrr}-\frac{1}{6} & 0 & 0 \\ 0 & \frac{1}{12} & 0 \\ 0 & 0 & -\frac{1}{8}\end{array}\right] \).
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Mail went to buy some veg he bought x kgs of tomato and y kgs of potato the total cost of veg comes out to be rs 200 now if the cost of 1 kg of tomato is rs50 and 1 kg of potato rs20 then ans the follow (1) liner equation to represent the total cost (2) if mail bought x kg of tomato and 2.5 kg of potato find the value of x (3) find the point at which the graph of 5x+2y=20 cuts x axis
Answer:
Linear equation to represent the total cost:
Let x be the number of kgs of tomatoes Mail bought, and y be the number of kgs of potatoes Mail bought. The cost of x kgs of tomato at Rs. 50 per kg is 50x, and the cost of y kgs of potato at Rs. 20 per kg is 20y. Therefore, the total cost of the vegetables is:
Total cost = 50x + 20y
Substituting the value of total cost as Rs. 200, we get:
50x + 20y = 200
This is the required linear equation to represent the total cost.
Finding the value of x:
Let's assume that Mail bought x kgs of tomato and 2.5 kgs of potato. Using the equation derived above:
50x + 20(2.5) = 200
Simplifying the equation:
50x + 50 = 200
50x = 150
x = 3
Therefore, Mail bought 3 kgs of tomato.
Finding the point at which the graph of 5x+2y=20 cuts x-axis:
To find the point at which the graph of 5x+2y=20 cuts the x-axis, we need to set y = 0 in the equation and solve for x:
5x + 2(0) = 20
5x = 20
x = 4
Therefore, the point where the graph of 5x+2y=20 cuts the x-axis is (4,0).
A solid cylinder is cut vertically through its center. Its radius and height are 6 cm and 15 cm, respectively. What is the area of the resulting shape?
The area of the resulting shape is 90 sq meters
How to determine the area of the resulting shape?The cylinder represents the given parameter
Such that we have the following dimensions
Radius = 6 cm
Height = 15 cm
When the solid cylinder is cut vertically through its center, we have a rectangle with the following dimensions
Length = 6 cm
Width = 15 cm
The area is then calculated as
Area = Length * Width
So, we have
Area = 6 * 15
Evaluate
Area = 90
Hence, the area is 90 square meters
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