Answers
i think the answer is 4 hope it helps
Related Questions
A hotel has 210 units. All rooms are occupied when the hotel charges $90 per day for a room. For every increase of x dollars in the daily room rate, there are x rooms vacant. Each occupied room costs $24 per day to service and maintain. What should the hotel charge per day in order to maximize daily profit
Answers
Answer: [tex]\$162[/tex]
Step-by-step explanation:
Given
There are 210 units of hotel
Each room charge $90 when fully occupied
occupied room charges is $24 per day
Suppose x rooms are vacant, so, there 210-x occupied
Daily expense on maintenance[tex]=(210-x)\times 24[/tex]
Earning from 210-x rooms[tex]=(210-x)(90+x)[/tex]
Profit(P)=Earning - Expense
[tex]\Rightarrow P=(210-x)(90+x)-(210-x)24\\\Rightarrow P=21,600+210x-90x-x^2-5040+24x\\\Rightarrow P=-x^2+144x+16,560\\[/tex]
To get the maximum profit, determine the derivative of P w.r.t. x and equate it to zero
[tex]\Rightarrow \dfrac{dP}{dx}=-2x+144\\\\\Rightarrow -2x+144=0\\\\\Rightarrow x=72[/tex]
Charges for maximum profit is [tex]90+72=\$162[/tex]
Number of occupied rooms are [tex]210-72=138[/tex]
The hotel charge per day will be $162 in order to maximize daily profit.
Given that, Hotel has 210 units
Let us consider that x rooms are vacant.
Total occupied rooms = [tex](210-x)[/tex]
Since, Each occupied room costs $24 per day to service and maintain
So, Total expense = [tex]24*(210-x)=5040-24x[/tex]
Since, For all rooms are occupied when the hotel charges $90 per day for a room. For every increase of x dollars in the daily room rate, there are x rooms vacant.
So, Total earing = [tex](210-x)*(90+x)=-x^{2} +120x+21600[/tex]
Profit = Total earning - Total expense
[tex]P=-x^{2} +120x+21600-5040+24x\\\\P=-x^{2} +144x+16560[/tex]
In order to maximize profit, Differentiate profit expression with respect to x.
[tex]\frac{dP}{dx}=\frac{d}{dx}(-x^{2} +144x+16560) \\\\\frac{dP}{dx}=-2x+144=0\\\\\\x=\frac{144}{2}=72[/tex]
Thus, Hotel charges per day for maximum profit = [tex]90+x=90+72=162[/tex]
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Match each sequence below to statement that BEST fits it.
Z. The sequence converges to zero;
I. The sequence diverges to infinity;
F. The sequence has a finite non-zero limit;
D. The sequence diverges.
_______ 1. ns in (1/n)
_______2. ln(ln(ln(n)))
_______3. (ln(n))/n
_______4. n!/n^1000
Answers
Answer: hello your question is poorly written attached below is the complete question
answer:
1 ) = I (
2) = F
3) = Z
4) = D
Step-by-step explanation:
attached below is the required solution.
1 ) = I ( The sequence diverges to infinity )
2) = F ( The sequence has a finite non-zero limit )
3) = Z ( The sequence converges to zero )
4) = D ( The sequence diverges )
Find the missing segment in the image below
Answers
Answer:
x = 12
Step-by-step explanation:
Missing length of the segment is the altitude of the right triangle.
Based on the geometric mean theorem, we would have the following:
h = √(ab)
Where,
h = x
a = 16
b = 9
Plug in the values:
x = √(16*9)
x = √144
x = 12
An architectural drawing lists the scale as 1/4" = 1'. If a bedroom measures 212" by 334" on the drawing, how large is the bedroom?
Answers
Answer:
53 by 83.5.
just divided the two numbers by 1/4
Answer:
848 and 1336
Step-by-step explanation:
You would actually multiply 212 and 334 by 4.
You need to multiply 1/4 by 4 to get 1.
212 x 4 = 848
334 x 4 =1336
What is 50g as a percentage of one kg?
Answers
Answer:
5 %
Step-by-step explanation:
1000 g = 1 kg
50 kg = 0.05 kg
0.05 = 5%
Therefore, 50 g as a percentage of 1kg is 5%.
At a large Midwestern university, a simple random sample of 100 entering freshmen in 1993 found that 20 of the sampled freshmen finished in the bottom third of their high school class. Admission standards at the university were tightened in 1995. In 1997, a simple random sample of 100 entering freshmen found that only 10 finished in the bottom third of their high school class. Let p 1 and p 2 be the proportions of all entering freshmen in 1993 and 1997, respectively, who graduated in the bottom third of their high school class. What is a 90% plus four confidence interval for p 1 – p 2?
Answers
Answer:
The 90% confidence interval for the difference of proportions is (0.01775,0.18225).
Step-by-step explanation:
Before building the confidence interval, we need to understand the central limit theorem and subtraction of normal variables.
Central Limit Theorem
The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
Subtraction between normal variables:
When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.
p1 -> 1993
20 out of 100, so:
[tex]p_1 = \frac{20}{100} = 0.2[/tex]
[tex]s_1 = \sqrt{\frac{0.2*0.8}{100}} = 0.04[/tex]
p2 -> 1997
10 out of 100, so:
[tex]p_2 = \frac{10}{100} = 0.1[/tex]
[tex]s_2 = \sqrt{\frac{0.1*0.9}{100}} = 0.03[/tex]
Distribution of p1 – p2:
[tex]p = p_1 - p_2 = 0.2 - 0.1 = 0.1[/tex]
[tex]s = \sqrt{s_1^2+s_2^2} = \sqrt{0.04^2 + 0.03^2} = 0.05[/tex]
Confidence interval:
[tex]p \pm zs[/tex]
In which
z is the z-score that has a p-value of [tex]1 - \frac{\alpha}{2}[/tex].
90% confidence level
So [tex]\alpha = 0.1[/tex], z is the value of Z that has a p-value of [tex]1 - \frac{0.1}{2} = 0.95[/tex], so [tex]Z = 1.645[/tex].
The lower bound of the interval is:
[tex]p - zs = 0.1 - 1.645*0.05 = 0.01775 [/tex]
The upper bound of the interval is:
[tex]p + zs = 0.1 + 1.645*0.05 = 0.18225 [/tex]
The 90% confidence interval for the difference of proportions is (0.01775,0.18225).
Hello people can you please help me on this I've been stuck on it for like 30 minuets now
Answers
Answer:
Step 1: Complete the first equation
0.01 is a hundredth, therefore if we have 1.86 then we have 186 hundredths.
Step 2: Complete the second equation
1.86 / 2 = 0.93
0.01 is a hundredth, therefore if we have 0.93 then we have 93 hundredths.
Step 3: Complete the third equation
1.86 / 2 = 0.93
Which choices are equivalent to the quotient below? Check All That Apply.
Answers
Answer:
Option C. √5/3
Step-by-step explanation:
We'll begin by simplifying √15 / 3√3. This can be obtained as follow:
√15 / 3√3
Rationalise
√15 / 3√3 × (√3 /√3)
(√15 × √3) / (3√3 × √3)
√45 / (3 × 3)
√45 / 9
Recall:
√45 = √(9 × 5) = √9 × √5 = 3√5
Thus,
√45 / 9 = 3√5 / 9
√45 / 9 = √5 / 3
Therefore,
√15 / 3√3 = √5 / 3
Thus, option C gives the right answer to the question.
Pls help
Minni is arranging 3 different music CDs in a row on a shelf. Create a sample space for the arrangement of a jazz CD (J), a pop CD (P), and a rock CD (R).
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Given:
Minni is arranging 3 different music CDs in a row on a shelf.
To find:
The sample space for the arrangement of a jazz CD (J), a pop CD (P), and a rock CD (R).
Solution:
We know that the total number of ways to arrange n items is n!.
Minni is arranging 3 different music CDs in a row on a shelf. So, the total number of ways is:
[tex]3!=3\times 2\times 1[/tex]
[tex]3!=6[/tex]
The sample space for the arrangement of a jazz CD (J), a pop CD (P), and a rock CD (R) is:
{JPR, JRP, PJR, PRJ, RJP, RPJ}
Therefore, the required sample space is {JPR, JRP, PJR, PRJ, RJP, RPJ}.
Answer:
RPJ, JPR, PRJ, JRP, RJP, JPR.
Step-by-step explanation:
I need help with this question
Answers
Answer:
7
This question is tying to introduce an idea that
eventually becomes a calculus question ....
(4^2 + 3(4)) - (0^2+3(0))
4 - 0
28 - 0 = 7
4
Step-by-step explanation:
Answer:
7
Step-by-step explanation:
1. Approach
The average rate of change can simply be defined as the slope of a line that passes through any two points on a coordinate plane. In this situation, one is given a function, and one is asked to find the rate of change over an interval.
Given function: [tex]f(x)=x^2+3x[/tex]
Intervale, [tex][0, 4][/tex]
This can be done by evaluating the endpoints of the interval by substituting them into the function. Then writing the resulting the form of a point on the coordinate plane ([tex]x, y[/tex]). Finally, one can find the slope of the line that passes through the points by using the following slope formula,
[tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
Where ([tex]x_1,y_1[/tex]) and ([tex]x_2,y_2[/tex]) are coordinate points.
2. Find the points
With the function (f(x)), substitute the end points of the itnerval ( [0, 4] ) into the function to generate coordinate points,
[tex]f(x)=x^2+3x[/tex]
[ 0, 4 ]
[tex]f(0)=(0)^2+3(0)\\=0 + 0\\ =0[/tex]
Point: (0, 0)
[tex]f(4)=(4)^2+3(4)\\= 16 + 12\\ = 28[/tex]
Point: (4, 28)
3. Find the average rate of change,
Now substitute the points the formula to find the slope, then simplify to evaluate
[tex]\frac{y_2-y_1}{x_2-x_1}[/tex]
(0, 0), (4, 28)
Substitute,
[tex]\frac{y_2-y_1}{x_2-x_1}\\\\=\frac{28-0}{4-0}\\\\=\frac{28}{4}\\\\= 7[/tex]
The average rate of change is 7.
Use the slope-intercept form of the linear equation to write an equation of the line with given slope and y-intercept.
Slope: -6/5 y intercept (0,8)
Answers
Answer:
5y + 6x = 40
Step-by-step explanation:
hope it is well understood?
m is the slope
b is the y intercept (0,y)
y=-6/5x+8
Look at images below. : ]
Answers
Answer:
1) A
B) 5.818 stops
Step-by-step explanation:
Number One is less than or equal to 21 because the person only has 21 dollars, so she can't spend more than 21.
B can be solved through the equation by first subtracting $5, and then dividing 2.75 by 16.
The function sin 0 is reciprocal of cot 0.
True
False
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Answer: False
Step-by-step explanation:
This is because sin 0 is equal to 0 but cot 0 is equal to undefined
You flip a coin that is not fair, the prbability of heads on each flip is 0.7. if the coin shows heads, you draw a marble from urn h with 1 blue and 4 red marbles. if the coin shows tails, you draw a marble from urn t with 3 blue and 1 red marble. Find the following probabilities:
a. The probability of choosing a red marble.
b. The probability of choosing a blue marble, given that the coin showed heads.
c. The probability that the coin showed tails, given that the marble was red.
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Solution :
P(H) = 0.7 ; P(T) = 0.3
If heads, then Urn H, 1 blue and 4 red marbles.
If tails, then Urn T , 3 blue and 1 red marbles.
a).
P ( choosing a Red marble )
= P (H) x P( Red from Urn H) + P (T) x P( Red from Urn T)
[tex]$=0.7 \times \frac{4}{5} + 0.3 \times \frac{1}{4}$[/tex]
= 0.56 + 0.075
= 0.635
b). If P (B, if coin showed heads)
If heads, then marble is picked from Urn H.
Therefore,
P (Blue) [tex]$=\frac{1}{5}$[/tex]
= 0.2
c). P (Tails, if marble was red)
[tex]$=P (T/R) = \frac{P(R/T)}{P(R)} \ P(T)$[/tex]
Where P (R/T) = P ( red, if coin showed tails)
[tex]$=\frac{1}{4}$[/tex]
= 0.25 (As Urn T is chosen)
P (R) = P (Red) = 0.635 (from part (a) )
P (T) = P (Tails) = 0.3
∴ [tex]$P(T/R) = \frac{0.25 \times 0.3}{0.635}$[/tex]
= 0.118
There are 165 children taking swimming lessons at the pool. If 10 children will be assigned to each instructor, how many instructors are needed?
Answers
Answer:
17 instructors
Step-by-step explanation:
If each instructor will get 10 children, we have to divided the total number of children taking swimming lessons by the number of children assigned to each instructor (10):
165/10 = 16.5
Unfortunately, we can't have 16 and a half instructors. Since 5 children are remaining, we can round up 16.5 to 17 and get 17 instructors. This implies that 16 instructors will teach 10 children (160 in total) and 1 instructor will teach 5 children (5 in total). 160+5 = 165 total children.
Use the graph of ƒ to find ƒ(2).
0.5
–8
–0.5
Does not exist
Answers
Answer:
Step-by-step explanation:
When you're looking to find things like f(2) and f(4) and f(-3000), etc. the number inside the parenthesis is an x value. Look to the graph, find that x value, and locate the y value that corresponds to it. f(2) = -8. f(-1) = 4. f(1) = -4. See?
Answer:
does not exist
Step-by-step explanation:
that what i put hope it helps
Drag each equation to the correct location on the table.
Match the equations with the value of x that makes them true.
5x - 2x - 4 = 5
5x - (3x - 1) = 7
x + 2x + 3 = 9
2(2x - 3) = 6
4x - (2x + 1) = 3
5(x + 3) = 25
x = 2
X = 3
Answers
Answer:
The answer to your questions are given below.
Step-by-step explanation:
To answer the question given above, we shall determine the value of x in each equation. This can be obtained as follow:
5x - 2x - 4 = 5
3x - 4 = 5
Collect like terms
3x = 5 + 4
3x = 9
Divide both side by 3
x = 9/3
x = 3
5x - (3x - 1) = 7
Clear the bracket
5x - 3x + 1 = 7
2x + 1 = 7
Collect like terms
2x = 7 - 1
2x = 6
Divide both side by 2
x = 6/2
x = 3
x + 2x + 3 = 9
3x + 3 = 9
Collect like terms
3x = 9 - 3
3x = 6
Divide both side by 3
x = 6/3
x = 2
2(2x - 3) = 6
Clear the bracket
4x - 6 = 6
Collect like terms
4x = 6 + 6
4x = 12
Divide both side by 4
x = 12/4
x = 3
4x - (2x + 1) = 3
Clear the bracket
4x - 2x - 1 = 3
2x - 1 = 3
Collect like terms
2x = 3 + 1
2x = 4
Divide both side by 2
x = 4/2
x = 2
5(x + 3) = 25
Clear the bracket
5x + 15 = 25
Collect like terms
5x = 25 - 15
5x = 10
Divide both side by 5
x = 10/5
x = 2
SUMMARY:
x = 2
x + 2x + 3 = 9
4x - (2x + 1) = 3
5(x + 3) = 25
x = 3
5x - 2x - 4 = 5
5x - (3x - 1) = 7
2(2x - 3) = 6
Alejandro wants to adopt a puppy from an animal shelter. At the shelter, he finds eight puppies that he likes: a male and female puppy from each of the four breeds of and Labrador. The puppies are each so cute that Alejandro cannot make up his mind, so he decides to pick the dog randomly. Find the probability that Alejandro chooses a .
Answers
Answer:
Hence the required probability is, 3/4
Step-by-step explanation:
At the shelter, he likes :
a male coolie, a female coolie, a male boxer, a female boxer, a male beagle, a female beagle, a male Labrador, and a female Labrador.
Let, A denote the event of selecting a male coolie and B denote the event of selecting a male Labrador.
P(A) = 1/8 = P(B)
Here the probability of selecting a puppy except A & B is,
P(AUB)c = 1 - P(AUB) = 1 - { P(A) + P(B) } = 1 - 1/8 - 1/8 = 3/4
How much would $200 invested at 5% interest compounded monthly be
worth after 9 years?
Answers
9514 1404 393
Answer:
$313.37
Step-by-step explanation:
The compound interest formula is used to find that value.
A = P(1 +r/12)^(12t)
P compounded monthly at annual rate r for t years.
A = $200(1 +0.05/12)^(12·9) ≈ $313.37
Can someone please help!!! Sec. 12.7 #78
Answers
N is one of the numbers below. N is such that when multiplied by 0.75 gives 1. Which number is equal to
N?
A) 1 1/2
B 1 1/3
C) 5/3
D) 3/2
Answers
Answer:
it should be letter c 5/3 I could be wrong but I hope this help
What is the equation of exponential regression equation? Round all numbers you your answer to three decimal places
Answers
Given:
The given values are:
[tex]a=0.2094539899[/tex]
[tex]b=2.507467975[/tex]
[tex]r^2=0.9435996398[/tex]
[tex]r=0.9713905701[/tex]
To find:
The exponential regression equation for the given values (Rounded to three decimal places).
Solution:
The general form of exponential regression equation is:
[tex]y=a\cdot b^x[/tex] ...(i)
Where, a is the initial value and b is the growth/decay factor.
We have,
[tex]a=0.2094539899[/tex]
[tex]b=2.507467975[/tex]
Round these numbers to three decimal places.
[tex]a\approx 0.209[/tex]
[tex]b\approx 2.507[/tex]
Substitute [tex]a=0.209, b=2.507[/tex] in (i) to find the exponential regression equation.
[tex]\hat{y}=0.209\cdot 2.507^x[/tex]
Therefore, the correct option is C.
rewrite -4<x<-1 using absolute value sign
Answers
[tex] - | 4 | < x < - |1| [/tex]
dunno if that's the desired form tough, but it states the same definition
The given inequality rewritten using absolute value sign as |-4|<x<|-1|.
What are inequalities?Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.
The given inequality is -4<x<-1.
An absolute value inequality is an expression with absolute functions as well as inequality signs.
Here, using absolute value sign we get
|-4|<x<|-1|
Therefore, the given inequality rewritten using absolute value sign as |-4|<x<|-1|.
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What is the sum of 2 and 3 subtracted from the product of 2 and the difference of 7 and 4? The answer is 1, but how is it solved?
Answers
Answer: -1
Step-by-step explanation:
sum of 2 and 3 subtracted from the product of 2 difference of 7 and 4
(2+3) - 2 ( 7 - 4 ) = -1
(2+3)-2(7-4) =
5 - 2(3) =
5-6 = -1
The sum of 2 and 3 subtracted from the product of 2 and the difference of 7 and 4 is equivalent to 1.
What is Equation Modelling?Equation modelling is the process of writing a mathematical verbal expression in the form of a mathematical expression for correct analysis, observations and results of the given problem.
We have the sum of 2 and 3 subtracted from the product of 2 and the difference of 7 and 4
From the question, we can model the equation as -
x = 2 × (7 - 4) - (2 + 3)
x = 2(3) - 5
x = 6 - 5
x = 1
Therefore, the sum of 2 and 3 subtracted from the product of 2 and the difference of 7 and 4 is equivalent to 1.
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find the missing side length below brainly
Answers
9514 1404 393
Answer:
24
Step-by-step explanation:
Corresponding lengths are proportional. The ratio of the full length to the given part is ...
(30 +x)/30 = 45/25
1 +x/30 = 1 +4/5 . . . perform the division
x/30 = 4/5 . . . . . . . . subtract 1
x = 30(4/5) = 24
The missing side length is 24.
a-bcosc/c-bcosa=sinc/sina
Answers
=ac-b(cosc-ccosa)
=sinc/sina
PLEASE HELP
Identify the first five terms of the sequence in which a, = 3n2 - 1.
Answers
Step-by-step explanation:
you cannot just put the actual numbers in and calculate ?
and you can't provide the correct problem statement, as it seems.
I assume you mean
an = 3n² - 1
a sequence starts with a1, so, n>=1
a1 = 3×1² - 1 = 3-1 = 2
a2 = 3×2² -1 = 3×4 - 1 = 12 - 1 = 11
a3 = 3×3² - 1 = 3×9 - 1 = 27 - 1 = 26
a4 = 3×4² - 1 = 3×16 - 1 = 48 - 1 = 47
a5 = 3×5² - 1 = 3×25 - 1 = 75 - 1 = 74
there, that is all there is to it. you really needed help with that ?
Dr. Kingston predicted that swearing can help reduce pain. In the study, each participant was asked to plunge a hand into icy water and keep it there as long as the pain would allow. In one condition, the participants repeatedly yelled their favorite curse words while their hands were in the water. In the other condition the participants repeated a neutral word. The table below presents the amount of time that participants kept their hand in the ice in each condition.
Swear Words
Neutral Words
98
56
70
61
52
47
87
60
46
32
120
92
72
53
41
31
1. Calculate the mean for the Swear Words condition:_______________
Answers
Answer:
Step-by-step explanation:
First, we add them all up.
98+70+52+87+46+120+72+41 = 586
Now, we divide 586 by the number of things there are. 586 / 8 = 73.25.
The mean of the swear words condition is 73.25.
Use the given information to find the number of degrees of freedom, the critical values χ2L and χ2R, and the confidence interval estimate of σ. It is reasonable to assume that a simple random sample has been selected from a population with a normal distribution. Nicotine in menthol cigarettes 99% confidence; n=23, s=0.28 mg.
df = (Type a whole number.)
χ2L = (Round to three decimal places as needed.)
χ2R = (Round to three decimal places as needed.)
The confidence interval estimate of σ is __ mg < σ < __ mg. (Round to two decimal places as needed.)
Answers
Answer:
χ²R = 8.643
χ²L = 42.796
0.20 < σ < 0.45
Step-by-step explanation:
Given :
Sample size, n = 23
The degree of freedom, df = n - 1 = 23 - 1 = 22
At α - level = 99%
For χ²R ; 1 - (1 - 0.99)/2= 0.995 ; df = 22 ; χ²R = 8.643
For χ²L ; (1 - 0.99)/2 = 0.005 ; df = 22 ; χ²L = 42.796
The confidence interval of σ ;
s * √[(n-1)/χ²L] < σ < s * √[(n-1)/χ²R)]
0.28 * √(22/42.796) < σ < 0.28 * √(22/8.643)
0.2008 < σ < 0.4467
0.20 < σ < 0.45
♥️♥️♥️♥️♥️♥️♥️♥️♥️ help me
Answers
9514 1404 393
Answer:
AC = 2.0 mm = 41.3 kgStep-by-step explanation:
The sum of torques about the pivot point is zero when the system is in equilibrium. That means the total of clockwise torques is equal to the total of counterclockwise torques. For this purpose, torque can be modeled by the product of mass and its distance from the pivot. The uniform beam can be modeled as a point mass at its center.
__
a) Let E represent the location of the center of mass of the beam. So, AE = 1.5 m. Then the distance from C to E is AC-AE = AC -1.5 and the CCW torque due to the beam's mass is (16 kg)(AC -1.5 m).
The distance from B to C is 3 m - AC, so the CW torque due to the particle at B is (7 kg)(3 -AC m)
These are equal, so we have ...
16(AC -1.5) = 7(3 -AC)
16AC -24 = 21 -7AC . . . . . eliminate parentheses
23AC = 45 . . . . . . . . . . . add 7AC+24
AC = 45/23 ≈ 1.957 . . divide by the coefficient of AC
AC ≈ 2.0 meters . . . . rounded to 1 dp
__
b) The torques in this scenario are ...
M(0.7) = 16(0.8) +7(2.3) . . . . . . AD = 0.7 m, DE = 0.8 m, DB = 2.3 m
M = 28.9/0.7 ≈ 41.286 . . . . simplify, divide by the coefficient of M
M = 41.3 kg . . . . rounded to 1 dp
_____
Additional comment
Torque is actually the product of force and distance from the pivot. Here, the forces are all downward, and due to the acceleration of gravity. The gravitational constant multiplies each mass, so there is no harm in dividing the equation by that constant, leaving the sum of products of mass and distance.
