A 10-gal aquarium is 3 in. higher than it is wide. Its length is 21 in, and its volume is 2730 in^3. What are the height and width of the aquarium?
x(x+3)......x^2+3x
21(x^2+3x)....
21x^2+63x-2730.....
I am stuck.....I think I am doing it wrong. I have the answers but I don't know how to get there....Help!
Let x be the width. x = 3 is then the height.
21 x*(x+3) = 2730
x(x+3) = 130
x^2 +3x -130 = 0
(x-10)(x+13) = 0
The only meaningful root is x = 10.
To solve the problem, let's use the given information to set up an equation.
1. Start by defining the variables:
- Let the width of the aquarium be 'x' inches.
- The height of the aquarium is given as 3 inches more than the width, so it can be represented as 'x + 3' inches.
- The length of the aquarium is given as 21 inches.
2. Calculate the volume of the aquarium:
The volume of a rectangular prism (aquarium) can be calculated by multiplying the length, width, and height:
Volume = Length × Width × Height
2730 = 21 × x × (x + 3)
3. Simplify the equation and solve for 'x':
Expand 21 × x × (x + 3):
2730 = 21x^2 + 63x
4. Rearrange the equation to form a quadratic equation:
21x^2 + 63x - 2730 = 0
5. Solve the quadratic equation:
You can solve the equation by factoring, completing the square, or using the quadratic formula. In this case, factoring may not be straightforward, so let's use the quadratic formula.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 21, b = 63, and c = -2730. Substituting these values in the quadratic formula, we get:
x = (-63 ± √(63^2 - 4 * 21 * -2730)) / (2 * 21)
6. Simplify and evaluate the quadratic formula:
Simplify the expression inside the square root:
x = (-63 ± √(3969 + 228240)) / 42
x = (-63 ± √(232209)) / 42
Calculate the square root:
x = (-63 ± 481) / 42
7. Solve for both solutions of 'x':
Solution 1:
x = (-63 + 481) / 42
x = 418 / 42
x ≈ 9.95 inches
Solution 2:
x = (-63 - 481) / 42
x = -544 / 42
x ≈ -13 inches (Ignore the negative value since aquarium dimensions cannot be negative)
8. Calculate the height of the aquarium:
The height is given as 'x + 3':
Height = 9.95 + 3
Height ≈ 12.95 inches
Therefore, the width of the aquarium is approximately 9.95 inches, and the height is approximately 12.95 inches.
To solve this problem, let's break it down step by step.
Step 1: Define the variables
Let's define the width of the aquarium as x inches. According to the problem, the height is 3 inches more than the width, so the height would be (x + 3) inches.
Step 2: Calculate the volume
The volume of the aquarium is given as 2730 cubic inches. The formula to calculate the volume of a rectangular prism (such as an aquarium) is V = length × width × height. Given that the length is 21 inches, we can plug in the values and solve for x:
2730 = 21 × x × (x + 3)
Step 3: Simplify the equation
Multiply 21 by x and x + 3:
2730 = 21x^2 + 63x
Now we have a quadratic equation that we need to solve.
Step 4: Rearrange the equation
Move all the terms to one side to create a quadratic equation equal to zero:
21x^2 + 63x - 2730 = 0
Step 5: Factor or use the quadratic formula to solve
In order to solve the equation, we can either factor it or use the quadratic formula.
In this case, factoring might not be easy due to large coefficients. So, we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 21, b = 63, and c = -2730. Plugging these values into the formula, we get:
x = (-63 ± √(63^2 - 4×21×(-2730))) / (2×21)
Step 6: Solve for x
Let's calculate the value of x using the quadratic formula. We'll consider both the positive and negative solutions:
x = (-63 ± √(3969 + 226680)) / 42
x = (-63 ± √(230649)) / 42
Calculating the square root:
x ≈ (-63 ± 480.22) / 42
Now, let's consider both cases separately:
Case 1: (x ≈ (-63 + 480.22) / 42)
Simplifying:
x ≈ (417.22) / 42
x ≈ 9.93
Case 2: (x ≈ (-63 - 480.22) / 42)
Simplifying:
x ≈ (-543.22) / 42
x ≈ -12.93
Since we're dealing with dimensions, only positive values make sense. Therefore, we discard the negative solution, x ≈ -12.93.
Step 7: Calculate the height
Now that we have the width (x ≈ 9.93), we can calculate the height by adding 3 to it:
Height = x + 3 ≈ 9.93 + 3 ≈ 12.93
So, the approximate height of the aquarium is 12.93 inches, and the approximate width is 9.93 inches.