Problem 50 - Algebra A Combined Function

Question

If the area of a right-triangularly shaped sail is 20 square feet and its base is 5 feet, find the height of the sail. \(\left(A=\frac{1}{2} b h\right)\)

Step-by-Step Solution

Answer

The height of the sail is 8 feet.

Step 1: Understand the Given Information

We know that the area of a right triangle is given formula as \( A = \frac{1}{2} b h \). Here, the area \( A \) is 20 square feet, and the base \( b \) is 5 feet.

Step 2: Substitute Known Values into the Formula

Substitute the given values into the area formula: \( 20 = \frac{1}{2} \times 5 \times h \).

Step 3: Simplify the Equation

Simplify the equation by solving for the height \( h \). First, calculate \( \frac{1}{2} \times 5 = 2.5 \). This results in the equation \( 20 = 2.5h \).

Step 4: Solve for the Height

To solve for \( h \), divide both sides of the equation by 2.5: \( h = \frac{20}{2.5} \).

Step 5: Calculate the Height

Perform the division: \( h = 8 \). Hence, the height of the sail is 8 feet.

Key Concepts

Area Calculation

Calculating the area of a right triangle is a straightforward task once you know the necessary parts. For right triangles, the area can be calculated using the formula \[A = \frac{1}{2} b h\]Where:

• \( A \) represents the area of the triangle.
• \( b \) stands for the base.
• \( h \) is the height.

This formula is derived from the general triangle area formula \( A = \frac{1}{2} \times \text{base} \times \text{height} \). It simplifies the multiplication process because you only need to focus on the perpendicular sides of the right triangle.

Remember, the base and height must be perpendicular to each other, which is guaranteed in a right triangle by definition. When using this formula, just plug in your known values to find either the area, base, or height, depending on which two quantities you already have.

Height Determination

Determining the height in a right triangle when you know the area and base is just a matter of rearranging the area formula. Using our formula, \[A = \frac{1}{2} b h\]we can solve for the height by manipulating it algebraically.

First, multiply both sides of the equation by 2 to eliminate the fraction:\[2A = b h\]Then, divide both sides by the base \( b \):\[h = \frac{2A}{b}\]This rearrangement isolates \( h \), allowing you to substitute the values for the area and base to find the height. If you know that the base is, for example, 5 feet, and the area is 20 square feet, substitute these into the formula:\[h = \frac{2 \times 20}{5} = 8\]Thus, you find that the height of the triangle is 8 feet. This method works efficiently for any right triangle if you know the area and base length.

Triangle Formula

The triangle formula used in this context is especially useful for right triangles, making calculations easier due to the inherent perpendicular nature of its sides. The formula \( A = \frac{1}{2} b h \) is the backbone of calculating parts of right triangles, such as area, base, or height.

It's crucial to remember:

• The base \( b \) and height \( h \) must be perpendicular.
• For a right triangle, the base and height are simply the two legs.
• This simplicity comes from the 90-degree angle, ensuring base and height form the needed perpendicular pair.

Calculating these values gives a practical understanding of geometrical properties and insights into real-world applications, such as assessing the dimensions of sails, as the exercise suggests. By employing the correct formula, mathematical problems involving right triangles become manageable and less daunting.