A pen in the shape of an isosceles right triangle with legs of length x ft and hypotenuse of length h ft is to be built. If fencing costs \$5/ft for the legs and \$10/ft for the hypotenuse, write the total cost C of construction as a function of h.

Verified step by step guidance

1. Start by identifying the cost components: the cost for the legs and the cost for the hypotenuse. The legs cost \$5 per foot, and the hypotenuse costs \$10 per foot.
2. Since the triangle is isosceles and right-angled, the two legs are equal in length, each being x feet. Therefore, the cost for the legs is 2 * 5 * x = 10x dollars.
3. The cost for the hypotenuse, which is h feet long, is 10 * h dollars.
4. The total cost C of construction is the sum of the costs for the legs and the hypotenuse. Therefore, C = 10x + 10h.
5. To express C as a function of h, use the Pythagorean theorem for the isosceles right triangle: x^2 + x^2 = h^2, which simplifies to 2x^2 = h^2. Solve for x in terms of h: x = sqrt(h^2/2). Substitute this expression for x into the cost function C = 10x + 10h to express C solely in terms of h.

Key Concepts

Isosceles Right Triangle Properties

An isosceles right triangle has two equal legs and a right angle between them. The relationship between the legs (x) and the hypotenuse (h) is defined by the Pythagorean theorem, where h = x√2. Understanding this relationship is crucial for expressing the cost function in terms of the hypotenuse.

Cost Function

A cost function represents the total cost associated with producing a certain quantity of goods or services. In this scenario, the cost function C is determined by the lengths of the triangle's sides and their respective costs: $5/ft for the legs and $10/ft for the hypotenuse. This function will be expressed in terms of h, the hypotenuse.

Substitution in Functions

Substitution is a method used in algebra to replace a variable with another expression. In this problem, we need to express the lengths of the legs in terms of the hypotenuse h. By substituting x = h/√2 into the cost function, we can derive a total cost function solely in terms of h.