Answer:
To solve for the dimensions of the rectangular poster, we'll use the information given in the question.
Let's assume the width of the poster is "w" inches. According to the question, the length is 10 more inches than three times the width. So, the length can be represented as "3w + 10" inches.
The area of a rectangle can be found by multiplying its length and width. In this case, the area is given as 99 square inches.
So, we can set up the equation:
w * (3w + 10) = 99
To solve this equation, we'll first expand the expression:
3w^2 + 10w = 99
Next, we'll rearrange the equation to get a quadratic equation in standard form:
3w^2 + 10w - 99 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, factoring is not straightforward, so let's use the quadratic formula:
w = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 3, b = 10, and c = -99.
Substituting these values into the quadratic formula, we get:
w = (-10 ± √(10^2 - 4 * 3 * -99)) / (2 * 3)
Simplifying further:
w = (-10 ± √(100 + 1188)) / 6
w = (-10 ± √1288) / 6
Now, we can simplify the square root:
w = (-10 ± √(4 * 322)) / 6
w = (-10 ± 2√322) / 6
w = (-5 ± √322) / 3
So, we have two possible solutions for the width of the poster: (-5 + √322) / 3 and (-5 - √322) / 3.
To find the length, we can substitute these values into the expression "3w + 10".
Let's calculate the values:
For the width w = (-5 + √322) / 3:
Length = 3w + 10 = 3 * ((-5 + √322) / 3) + 10 = -5 + √322 + 10 = √322 + 5
For the width w = (-5 - √322) / 3:
Length = 3w + 10 = 3 * ((-5 - √322) / 3) + 10 = -5 - √322 + 10 = -√322 + 5
Therefore, the dimensions of the rectangular poster are:
Width = (-5 + √322) / 3, Length = √322 + 5
or
Width = (-5 - √322) / 3, Length = -√322 + 5
Explanation: