The dimension of a rectangular metal box are 3 cm,5cm and 8cm.if the first two dimensions are increased by the same number of centimeters, while the third dimension remains the same, the new volume is 34 cubic cetimeters more than the original volume.what is the new dimension of the enlarged rectangular metal box?
(3+x)*(5+x)*8 = 120+34.
(3x+15 + x^2+5x)*8 = 154,
(x^2+8x+15)*8 = 154,
8x^2 + 64x + 120 = 154,
8x^2 + 64x - 34 = 0,
Use Quad. Formula:
X = (-64 +- Sqrt(4096+1088))/16
X = (-64 +- 72)/16 = 0.50 cm, or X = - 8.50 cm.(Not valid).
Solution: X = 0.50 cm. added.
New Dimensions = 3.50 cm, 5.50 cm, and 8 cm.
The dimension of rectangular mental box of 3 cm 5 cm and 8 cm is the first two dimension of the increase by the same number of cm while the third dimension remain the same the new value volume and 34 cm cubic more than the original volume what is the new dimension of the in large rectangular metal box
Well, well, well! It seems like we have some mathematical mischief to solve here! Let's put on our thinking hats and get cracking!
The original volume of the rectangular metal box can be calculated by multiplying the three dimensions: 3 cm x 5 cm x 8 cm = 120 cubic cm.
Now, let's say we increase the first two dimensions by the same number of centimeters, but keep the third dimension the same. Let's call this magical number of centimeters "x".
The new dimensions would be:
First dimension: 3 cm + x
Second dimension: 5 cm + x
Third dimension: 8 cm
The new volume would then be:
(3 cm + x) * (5 cm + x) * 8 cm
Now, we know that this new volume is 34 cubic centimeters more than the original volume, so we can set up the equation:
(3 cm + x) * (5 cm + x) * 8 cm = 120 cm^3 + 34 cm^3
Let's do some juggling with the numbers to solve for "x":
8(15 + 3x + 5x + x^2) = 154
120 + 16x + 8x + 8x^2 = 154
8x^2 + 24x + 120 - 154 = 0
8x^2 + 24x - 34 = 0
Now, we can use some clown magic called the quadratic formula to solve for "x". So here we go!
x = (-b ± √(b^2 - 4ac)) / (2a)
Using the quadratic formula with a = 8, b = 24, and c = -34, we find two solutions for "x". Let's ignore the negative solution, because we're dealing with dimensions and we can't have negative lengths. So, we take the positive solution:
x ≈ 0.752 cm
Now, to determine the new dimensions of the enlarged rectangular metal box, we add this "x" to the first two dimensions:
First dimension: 3cm + 0.752cm ≈ 3.752 cm
Second dimension: 5cm + 0.752cm ≈ 5.752 cm
Third dimension: 8 cm (unchanged)
Therefore, the new dimensions of the enlarged rectangular metal box are approximately 3.752 cm, 5.752 cm, and 8 cm. Voila!
Remember, math can sometimes be a circus act, but with a little humor and some clowning around, we can solve any problem!
Let's start by finding the original volume of the metal box.
Given dimensions:
Length = 3 cm
Width = 5 cm
Height = 8 cm
Original volume (V1) = Length x Width x Height
= 3 cm x 5 cm x 8 cm
= 120 cubic cm
Now, let's assume the increase in the first two dimensions by 'x' cm. In that case, the new dimensions would be:
Length = (3 + x) cm
Width = (5 + x) cm
Height = 8 cm
New volume (V2) = (Length x Width x Height) + 34
= (3 + x) cm x (5 + x) cm x 8 cm + 34
According to the given condition, the new volume is 34 cubic cm more than the original volume.
Therefore, we can set up the following equation:
(3 + x) cm x (5 + x) cm x 8 cm + 34 = 120 cubic cm
Now, let's solve this equation to find the value of 'x'.
To solve this problem, we need to use the formula for the volume of a rectangular box, which is V = L × W × H, where V is the volume, L is the length, W is the width, and H is the height.
Given that the original dimensions are 3 cm, 5 cm, and 8 cm, we can calculate the original volume by substituting these values into the formula:
Original volume = 3 cm × 5 cm × 8 cm = 120 cubic cm
Let's assume that the increased dimensions are x cm. Therefore, the new dimensions will be 3 cm + x cm, 5 cm + x cm, and 8 cm (as the third dimension remains the same). The new volume can be calculated in the same way:
New volume = (3 cm + x cm) × (5 cm + x cm) × 8 cm
According to the problem statement, the new volume is 34 cubic cm more than the original volume:
New volume = Original volume + 34 cubic cm
Therefore, we can set up the following equation:
(3 + x)(5 + x)(8) = 120 + 34
Now, let's solve this equation to find the value of x, which represents the amount by which the first two dimensions are increased.
(3 + x)(5 + x)(8) = 154
Expanding the left side:
(15 + 8x + 3x + x^2)(8) = 154
Simplifying further:
(15 + 11x + x^2)(8) = 154
120 + 88x + 8x^2 = 154
Rearranging the terms:
8x^2 + 88x + 120 - 154 = 0
8x^2 + 88x - 34 = 0
Dividing the equation by 2 to simplify it:
4x^2 + 44x - 17 = 0
Now we can solve this equation using factoring, the quadratic formula, or any other appropriate method.
By solving the quadratic equation, we can find the value of x, which represents the increase in dimensions. From there, we can determine the new dimensions of the rectangular metal box by adding x to the original dimensions (3 cm and 5 cm) while keeping the third dimension (8 cm) unchanged.