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Sagot :
Alright, let's work through this step-by-step to determine which statement is true.
1. Determine the height, [tex]\( h \)[/tex], of the tank:
The problem states that the height of the tank is one inch less than twice the width.
[tex]\[ h = 2w - 1 \][/tex]
2. Determine the length, [tex]\( l \)[/tex], of the tank:
It also states that the length of the tank is seven inches longer than the height.
[tex]\[ l = h + 7 \][/tex]
Substituting the expression we have for [tex]\( h \)[/tex]:
[tex]\[ l = (2w - 1) + 7 \][/tex]
Simplifying the expression for [tex]\( l \)[/tex]:
[tex]\[ l = 2w - 1 + 7 = 2w + 6 \][/tex]
So, the length [tex]\( l \)[/tex] is [tex]\( 2w + 6 \)[/tex].
3. Calculate the volume of the fish tank:
The volume, [tex]\( V \)[/tex], of a rectangular tank is given by the product of its length, width, and height.
[tex]\[ V = lwh \][/tex]
Substitute the expressions for [tex]\( l \)[/tex] and [tex]\( h \)[/tex]:
[tex]\[ V = (2w + 6) \cdot w \cdot (2w - 1) \][/tex]
Now expand and simplify this expression:
[tex]\[ V = (2w + 6) \cdot w \cdot (2w - 1) \][/tex]
First, distribute [tex]\( w \)[/tex] in the second part of the expression:
[tex]\[ V = (2w + 6)(2w^2 - w) \][/tex]
Then, expand the products:
[tex]\[ V = 2w \cdot 2w^2 - 2w \cdot w + 6 \cdot 2w^2 - 6 \cdot w \][/tex]
Simplify each term:
[tex]\[ V = 4w^3 - 2w^2 + 12w^2 - 6w \][/tex]
Combine like terms:
[tex]\[ V = 4w^3 + 10w^2 - 6w \][/tex]
4. Evaluate the given statements:
- Statement A: The trinomial expression [tex]\( 4w^3 - 10w^2 + 6w \)[/tex] represents the volume of the fish tank.
From our calculations, we have:
[tex]\[ V = 4w^3 + 10w^2 - 6w \][/tex]
So, statement A is not correct.
- Statement B: The binomial expression [tex]\( 2w + 6 \)[/tex] represents the length of the fish tank.
From our calculations, we found:
[tex]\[ l = 2w + 6 \][/tex]
This is true.
- Statement C: The binomial expression [tex]\( 4w^3 + 10w^2 \)[/tex] represents the volume of the fish tank.
From our calculations, we have:
[tex]\[ V = 4w^3 + 10w^2 - 6w \][/tex]
So, statement C is not correct.
- Statement D: The monomial expression [tex]\( 2w \)[/tex] represents the height of the fish tank.
From the given condition, the height [tex]\( h \)[/tex] is:
[tex]\[ h = 2w - 1 \][/tex]
So, statement D is not correct.
Based on our calculations and evaluation, the correct statement is:
B. The binomial expression [tex]\( 2w + 6 \)[/tex] represents the length of the fish tank.
1. Determine the height, [tex]\( h \)[/tex], of the tank:
The problem states that the height of the tank is one inch less than twice the width.
[tex]\[ h = 2w - 1 \][/tex]
2. Determine the length, [tex]\( l \)[/tex], of the tank:
It also states that the length of the tank is seven inches longer than the height.
[tex]\[ l = h + 7 \][/tex]
Substituting the expression we have for [tex]\( h \)[/tex]:
[tex]\[ l = (2w - 1) + 7 \][/tex]
Simplifying the expression for [tex]\( l \)[/tex]:
[tex]\[ l = 2w - 1 + 7 = 2w + 6 \][/tex]
So, the length [tex]\( l \)[/tex] is [tex]\( 2w + 6 \)[/tex].
3. Calculate the volume of the fish tank:
The volume, [tex]\( V \)[/tex], of a rectangular tank is given by the product of its length, width, and height.
[tex]\[ V = lwh \][/tex]
Substitute the expressions for [tex]\( l \)[/tex] and [tex]\( h \)[/tex]:
[tex]\[ V = (2w + 6) \cdot w \cdot (2w - 1) \][/tex]
Now expand and simplify this expression:
[tex]\[ V = (2w + 6) \cdot w \cdot (2w - 1) \][/tex]
First, distribute [tex]\( w \)[/tex] in the second part of the expression:
[tex]\[ V = (2w + 6)(2w^2 - w) \][/tex]
Then, expand the products:
[tex]\[ V = 2w \cdot 2w^2 - 2w \cdot w + 6 \cdot 2w^2 - 6 \cdot w \][/tex]
Simplify each term:
[tex]\[ V = 4w^3 - 2w^2 + 12w^2 - 6w \][/tex]
Combine like terms:
[tex]\[ V = 4w^3 + 10w^2 - 6w \][/tex]
4. Evaluate the given statements:
- Statement A: The trinomial expression [tex]\( 4w^3 - 10w^2 + 6w \)[/tex] represents the volume of the fish tank.
From our calculations, we have:
[tex]\[ V = 4w^3 + 10w^2 - 6w \][/tex]
So, statement A is not correct.
- Statement B: The binomial expression [tex]\( 2w + 6 \)[/tex] represents the length of the fish tank.
From our calculations, we found:
[tex]\[ l = 2w + 6 \][/tex]
This is true.
- Statement C: The binomial expression [tex]\( 4w^3 + 10w^2 \)[/tex] represents the volume of the fish tank.
From our calculations, we have:
[tex]\[ V = 4w^3 + 10w^2 - 6w \][/tex]
So, statement C is not correct.
- Statement D: The monomial expression [tex]\( 2w \)[/tex] represents the height of the fish tank.
From the given condition, the height [tex]\( h \)[/tex] is:
[tex]\[ h = 2w - 1 \][/tex]
So, statement D is not correct.
Based on our calculations and evaluation, the correct statement is:
B. The binomial expression [tex]\( 2w + 6 \)[/tex] represents the length of the fish tank.
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