Please show me how to set it up and solve:
You are making a rectangular table. The area of the table should be 10ft squared. You want the length of the table to be 1ft shorter than twice its width. What should the dimensions of the able be?
They should be 5 as the length and 2 as the width.
as for the setup:
let the width be w
so, the length is 2w-1
area = w(2w-1) = 10
2w^2 - w - 10 = 0
(2w-5)(w+2) = 0
w = 2.5
so, 2w-1 = 4
the table is 2.5 x 4
Note that Kelli set it up so length = 2w+1
To solve this problem, we can follow these steps:
Step 1: Assign variables
Let's assign variables to the dimensions of the table. We can call the width of the table "w" and the length of the table "l."
Step 2: Translate the given information into an equation
We know that the area of the table is 10ft², so we can set up an equation: A = l * w = 10
Step 3: Translate the given relationship between the length and width into an equation
The question states that the length of the table should be 1ft shorter than twice its width. Therefore, we can write the equation: l = 2w - 1
Step 4: Substitute the equation for length into the area equation
Now, we substitute 2w - 1 for l in the area equation: A = w * (2w - 1) = 10
Step 5: Solve the equation
To solve the equation, we need to simplify it and then solve for w. Expanding the equation, we have: 2w² - w = 10
Rearranging the equation, we have: 2w² - w - 10 = 0
Step 6: Factor or use the quadratic formula to solve the equation
To solve this quadratic equation, we can try to factor it or use the quadratic formula. However, factoring might not be immediately obvious, so let's use the quadratic formula:
w = (-b ± √(b² - 4ac)) / 2a
In this case, a = 2, b = -1, and c = -10. Substituting these values into the quadratic formula:
w = (-(-1) ± √((-1)² - 4 * 2 * -10)) / (2 * 2)
w = (1 ± √(1 + 80)) / 4
w = (1 ± √81) / 4
w = (1 ± 9) / 4
Simplifying, we have two possible solutions:
w₁ = (1 + 9) / 4 = 10 / 4 = 2.5
w₂ = (1 - 9) / 4 = -8 / 4 = -2
Since width cannot be negative, we discard the negative value for w.
Step 7: Calculate the length
To find the length, we use the equation l = 2w - 1:
l = 2 * 2.5 - 1 = 5 - 1 = 4
Therefore, the dimensions of the rectangular table should be a width of 2.5ft and a length of 4ft.