14. Question says that there is a triangle with a base of x units and height h units and two sides of length 10 units. Answer the following questions:
• Draw a representation of this situation (the triangle).
• Find a formula for h in terms of x. h = square root {10^2-(x/2)^2}
• What is the domain of this function?
• Graph this function over its domain.
• How does h change as x changes?
• Find a formula for the area of the triangle in terms of x only. Graph this function. ½*x*
• How does the area of the triangle change as x changes? Area increases
• For what value of x is the area of the triangle largest?
• For what value of h is the area of the triangle largest?
To answer these questions, let's go step by step:
1. Draw a representation of the situation:
We can draw a triangle with a base of length x units, two sides of length 10 units, and a height of h units. The height, h, represents a perpendicular line from the top vertex of the triangle to the base.
2. Find a formula for h in terms of x:
To find a formula for h, we can use the Pythagorean theorem. Since we have two sides of length 10 units, we can consider the triangle formed by the half of the base, the height, and one of the sides. Applying the Pythagorean theorem, we get:
(h)^2 + (x/2)^2 = (10)^2
Simplifying this equation, we can solve for h:
h = square root {10^2 - (x/2)^2}
3. Find the domain of the function:
The domain of the function represents all the possible values for x. Since the length of a triangle's base can be any positive value, the domain for this function is x > 0.
4. Graph the function over its domain:
To graph the function, you can plot points for different values of x in the domain and connect them to form a curve. Using a graphing tool or software can help visualize the graph accurately.
5. How does h change as x changes?
As x changes, the height of the triangle, h, will also change. You can observe this by looking at the graph of the function. Generally, as x increases, h will decrease, and as x decreases, h will increase.
6. Find a formula for the area of the triangle in terms of x only:
The formula for the area of a triangle is given by (1/2) * base * height. In this case, the base is x units, and the height is h units. Therefore, the formula for the area of the triangle in terms of x only is:
Area = (1/2) * x * h
7. Graph the area of the triangle as a function of x:
To graph the function for the area of the triangle, you can plot points using various values of x and calculate the corresponding areas. Connect these points to form a curve. You can also use a graphing tool or software to create an accurate graph.
8. How does the area of the triangle change as x changes?
As x changes, the area of the triangle will also change. From the formula, Area = (1/2) * x * h, we can see that the area increases linearly with x.
9. For what value of x is the area of the triangle largest?
To find the value of x for which the area of the triangle is largest, we need to maximize the area formula. Since the formula for the area includes x and h, we can substitute the expression for h found earlier:
Area = (1/2) * x * square root {10^2 - (x/2)^2}
To find the maximum area, you can use various methods like differential calculus or plotting the graph and finding the highest point.
10. For what value of h is the area of the triangle largest?
Similarly, to find the value of h for which the area of the triangle is largest, we need to maximize the area formula. We can substitute the expression for h found earlier into the area formula:
Area = (1/2) * x * h
Then, we can determine the maximum area by using methods like differential calculus or plotting the graph and finding the highest point.