A right triangle has one leg four times as long as the other. Find a function that models its perimeter P in terms of the length x of the shorter leg.
Well, it seems like the right triangle forgot to apply for a leg extension. But fear not, I shall do some mathematical clownery to solve this for you!
Let's call the length of the shorter leg "x". Since the other leg is four times as long as the shorter leg, we can say the length of the longer leg is 4x.
Now, let's calculate the hypotenuse of the right triangle using the Pythagorean theorem: a² + b² = c². In our case, a is x, b is 4x, and c is the hypotenuse.
So, x² + (4x)² = c²
x² + 16x² = c²
17x² = c²
Now, to find the perimeter (P) of the right triangle, we add up all the sides, which are x, 4x, and c:
P = x + 4x + c
Since c = √(17x²), we can substitute this value into the perimeter equation:
P = x + 4x + √(17x²)
Simplifying it further, we get:
P = 5x + √(17x²)
And there you have it! The function that models the perimeter (P) of the right triangle in terms of the length x of the shorter leg is P = 5x + √(17x²). However, I must warn you, this function has been known to cause occasional fits of laughter. Proceed with caution!
To find a function that models the perimeter of the right triangle, we need to determine the lengths of the other two sides.
Let x represent the length of the shorter leg. Since the longer leg is four times as long as the shorter leg, we can say that the longer leg has a length of 4x.
Using the Pythagorean theorem, the formula for the hypotenuse of a right triangle is given by:
c^2 = a^2 + b^2
Where c represents the hypotenuse and a and b represent the lengths of the legs.
In this case, x represents one of the legs (a or b) and 4x represents the other leg (a or b). So we can rewrite the formula as:
c^2 = x^2 + (4x)^2
Simplifying this equation:
c^2 = x^2 + 16x^2
c^2 = 17x^2
Taking the square root of both sides to solve for c:
c = sqrt(17x^2)
c = sqrt(17) * x
Now that we have the lengths of all three sides, we can determine the perimeter of the triangle:
P = x + 4x + sqrt(17) * x
P = 5x + sqrt(17) * x
Therefore, the function that models the perimeter of the right triangle in terms of the length x of the shorter leg is:
P(x) = 5x + sqrt(17) * x
To find a function that models the perimeter of the right triangle in terms of the length of the shorter leg, we need to understand the relationship between the triangle's sides.
Let's assume the shorter leg has a length of x, and the longer leg is four times as long, which means its length is 4x.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the hypotenuse is the square root of (x^2 + (4x)^2), as the shorter leg is x and the longer leg is 4x.
Now, we can calculate the perimeter of the triangle by adding the lengths of all three sides:
P = x + 4x + √(x^2 + (4x)^2)
Simplifying this, we get:
P = 5x + √(x^2 + 16x^2)
P = 5x + √(17x^2)
P = 5x + x√17
Thus, the function that models the perimeter P in terms of the length x of the shorter leg is:
P(x) = 5x + x√17