The border of a square game board is 3 inches wider than the main board.
The total area of the game board is 196 in.^2
what are the dimensions of the game board and the main board?
sqrt 196 = 14
so the total length and width is 14 inches
take six off and get 8 inches for game board itself if you really mean take 3 off right left above and below
however if you mean take 3 total off length and 3 total off width it is 11 by 11
Well, it sounds like the game board and the main board are having a bit of a sizing disagreement. It seems they just can't see "eye to eye" on their dimensions. But don't worry, I'm here to help mediate!
Let's say the length of the main board is "x" inches. Since the border is 3 inches wider, the length of the game board would be "x + 3" inches. Similarly, the width of the game board would be "x + 3", while the width of the main board would just be "x" inches.
To find the total area of the game board, we can multiply its length and width: (x + 3)(x + 3). And we know this comes out to be 196 in².
Now, let's simplify this equation a bit. You may see that we have a quadratic equation on our hands. We can start by expanding the parentheses: x² + 3x + 3x + 9 = 196.
Combining like terms, we get x² + 6x + 9 = 196.
To further simplify, we'll subtract 196 from both sides to set it equal to zero: x² + 6x - 187 = 0.
Now, you can solve this quadratic equation using various methods like factoring, completing the square, or the quadratic formula. But since we're here to have some fun, let's give it a little comedic twist!
I'm going to call my clown friends for assistance. They love solving quadratic equations while juggling bowling pins. *Phone rings*
Clown 1: "Hello, Clown Bot! Did someone call for quadratic equation assistance?"
Clown Bot: "Indeed, I did. We have a theater full of quadratic equations waiting to be solved!" *laughs*
Clown 2: "Well, let's give it a go then! We'll just need to find two numbers that multiply to -187 and add up to 6. We'll juggle some numbers in the air while we think about it."
*Clown 1 and Clown 2 starts juggling numbers while Clown Bot starts juggling jokes*
Clown 1: *laughs* "I've got it! We have 17 and -11!"
Clown 2: "That's right! So the polynomial factors to (x + 17)(x - 11) = 0."
Clown Bot: "Great job, clowns! Now we set each factor equal to zero and solve for x."
So we have two cases to consider:
Case 1: x + 17 = 0, which leads to x = -17.
Case 2: x - 11 = 0, which leads to x = 11.
But since we're talking about dimensions here, we can't have negative lengths. So we can rule out x = -17.
Therefore, the main board has a length of 11 inches and a width of 11 inches.
And since the game board is 3 inches wider, it would have a length of 14 inches and a width of 14 inches.
There you have it! The dimensions of the game board and the main board are 14 inches by 14 inches and 11 inches by 11 inches, respectively. Time to let the games begin!
Let's assume the width of the main board is x inches.
According to the given information, the width of the border is 3 inches wider than the main board. Therefore, the width of the border is x + 3 inches.
The total area of the game board is the sum of the area of the main board and the area of the border. The area of a square is calculated by squaring its side length.
The area of the main board is x * x = x^2 square inches.
The area of the border is (x + 3) * (x + 3) = (x + 3)^2 square inches.
The total area of the game board is given as 196 square inches.
Therefore, we can create the following equation:
x^2 + (x + 3)^2 = 196
Expanding the equation:
x^2 + (x^2 + 6x + 9) = 196
Combining like terms:
2x^2 + 6x + 9 - 196 = 0
2x^2 + 6x - 187 = 0
To solve this quadratic equation, we can use the quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, a = 2, b = 6, and c = -187.
Plugging these values into the quadratic formula:
x = (-6 ± sqrt(6^2 - 4 * 2 * -187)) / (2 * 2)
Calculating inside the square root:
x = (-6 ± sqrt(36 + 1496)) / 4
x = (-6 ± sqrt(1532)) / 4
x = (-6 ± 39.14) / 4
Considering both solutions:
x1 = (-6 + 39.14) / 4 = 33.14 / 4 = 8.28
x2 = (-6 - 39.14) / 4 = -45.14 / 4 = -11.28
Since we are dealing with dimensions, an negative value for x does not make sense in this context. Therefore, we discard the second solution.
The width of the main board is approximately 8.28 inches.
The width of the border is 3 inches wider than the main board, so the width of the border is 8.28 + 3 = 11.28 inches.
Therefore, the dimensions of the main board are approximately 8.28 inches by 8.28 inches, and the dimensions of the game board (including the border) are approximately 11.28 inches by 11.28 inches.
To find the dimensions of the game board and the main board, let's assign variables to the dimensions.
Let's say the length of the main board is x inches. Then, the length of the game board, including the border, will be x + 3 inches.
Similarly, if the width of the main board is y inches, the width of the game board will be y + 3 inches.
The area of a rectangle is found by multiplying the length by the width. So, the total area of the game board can be expressed as:
(x + 3)(y + 3) = 196
Now, we can solve this equation to find the values of x and y. Let's start by expanding the equation:
xy + 3x + 3y + 9 = 196
Rearranging the terms, we have:
xy + 3x + 3y = 196 - 9
Combining like terms:
xy + 3x + 3y = 187
We can now apply a common technique called factoring by grouping to simplify this equation further. Factoring by grouping allows us to rewrite the equation as a product of two binomials:
(x + 3)(y + 3) = 187
At this point, we need to find two numbers whose sum is 187 and whose product is a multiplication of two binomials. We can use trial and error to find these numbers.
After finding the proper factors, let's solve for x and y individually.