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If -10 < x < 0, 10 < 2y < 30 and 30 < 3z < 50, and x, y and z are integers, what is the difference between the maximum and minimum value that the expression x + 3y + 2z can take?
Correct answer is '46'. Can you explain this answer?
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Verified Answer
If -10 < x < 0, 10 < 2y < 30 and 30 < 3z < 50,...
Since x, y and z are integers, the ranges change to:
-10< x < 0, 5 < y < 15 and 10 < z < 16
Hence, the range for each term in the expression is:
- 10 < x < 0, 15 < 3 y < 45 and 20 < 2z < 32
Hence, the minimum value of x + 3y + 2z = -9 + 18 + 22 = 31
And, maximum value of x + 3y + 2z = 0 + 45 + 32 = 77
Required value = 77 - 31 = 46
Answer: 46
-10< x < 0, 5 < y < 15 and 10 < z < 16
Hence, the range for each term in the expression is:
- 10 < x < 0, 15 < 3 y < 45 and 20 < 2z < 32
Hence, the minimum value of x + 3y + 2z = -9 + 18 + 22 = 31
And, maximum value of x + 3y + 2z = 0 + 45 + 32 = 77
Required value = 77 - 31 = 46
Answer: 46
Most Upvoted Answer
If -10 < x < 0, 10 < 2y < 30 and 30 < 3z < 50,...
Given information:
-10x < />
10 + 2y = 30
30 + 3z = 50
Step 1: Solve the equations
From the first equation, we can conclude that x must be greater than 0.
From the second equation, we can solve for y:
10 + 2y = 30
2y = 30 - 10
2y = 20
y = 20/2
y = 10
From the third equation, we can solve for z:
30 + 3z = 50
3z = 50 - 30
3z = 20
z = 20/3
z ≈ 6.67
Since x, y, and z are integers, we round z to the nearest integer, which is 7.
Therefore, x = 1, y = 10, and z = 7.
Step 2: Calculate the expression x + 3y + 2z
x + 3y + 2z = 1 + 3(10) + 2(7)
= 1 + 30 + 14
= 45
Step 3: Find the maximum and minimum values of the expression
To find the maximum value, we need to maximize each variable:
- x is already at its maximum value of 1.
- y can be maximized by increasing it to the highest possible integer, which is 10.
- z can also be maximized by increasing it to the highest possible integer, which is 7.
Therefore, the maximum value of the expression is:
1 + 3(10) + 2(7) = 45
To find the minimum value, we need to minimize each variable:
- x is already at its minimum value of 1.
- y can be minimized by decreasing it to the lowest possible integer, which is -10.
- z can also be minimized by decreasing it to the lowest possible integer, which is -10.
Therefore, the minimum value of the expression is:
1 + 3(-10) + 2(-10) = 1 - 30 - 20 = -49
Step 4: Calculate the difference
The difference between the maximum and minimum values is:
45 - (-49) = 45 + 49 = 94
Therefore, the correct answer is 94.
-10x < />
10 + 2y = 30
30 + 3z = 50
Step 1: Solve the equations
From the first equation, we can conclude that x must be greater than 0.
From the second equation, we can solve for y:
10 + 2y = 30
2y = 30 - 10
2y = 20
y = 20/2
y = 10
From the third equation, we can solve for z:
30 + 3z = 50
3z = 50 - 30
3z = 20
z = 20/3
z ≈ 6.67
Since x, y, and z are integers, we round z to the nearest integer, which is 7.
Therefore, x = 1, y = 10, and z = 7.
Step 2: Calculate the expression x + 3y + 2z
x + 3y + 2z = 1 + 3(10) + 2(7)
= 1 + 30 + 14
= 45
Step 3: Find the maximum and minimum values of the expression
To find the maximum value, we need to maximize each variable:
- x is already at its maximum value of 1.
- y can be maximized by increasing it to the highest possible integer, which is 10.
- z can also be maximized by increasing it to the highest possible integer, which is 7.
Therefore, the maximum value of the expression is:
1 + 3(10) + 2(7) = 45
To find the minimum value, we need to minimize each variable:
- x is already at its minimum value of 1.
- y can be minimized by decreasing it to the lowest possible integer, which is -10.
- z can also be minimized by decreasing it to the lowest possible integer, which is -10.
Therefore, the minimum value of the expression is:
1 + 3(-10) + 2(-10) = 1 - 30 - 20 = -49
Step 4: Calculate the difference
The difference between the maximum and minimum values is:
45 - (-49) = 45 + 49 = 94
Therefore, the correct answer is 94.
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Community Answer
If -10 < x < 0, 10 < 2y < 30 and 30 < 3z < 50,...
The maximum and minimum integral values that x, y and z can take are x=-1, y=14, z=16 and x=-9, y=6, z=9 respectively. Substituting these values, we obtain the maximum and minimum value of x+3y+2z as 82 and 36 respectively. 82-36=46.
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If -10 < x < 0, 10 < 2y < 30 and 30 < 3z < 50, and x, y and z are integers, what is the difference between the maximum and minimum value that the expression x + 3y + 2z can take?Correct answer is '46'. Can you explain this answer?
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