Question

The table shows the functions representing the height and base of a triangle for different values of x.

Height
Base
f(x)=x² + 3
g(x) = 2x
4
1
7
4
3
12
6
4
19
8
The area of the triangle when x = 2 is 14. Which equation can be used to represent the area of the triangle, A(x)?

Answer 1

Option 2 is the correct answer.

Explanation:

The equation that can be used to represent the area of the triangle is:

A(x) = 0.5 * (f ⋅ g)(x)

Let's break it down step by step:

Step 1: Understanding the equation components

f(x) represents the height of the triangle, given as x² + 3.

g(x) represents the base of the triangle, given as 2x.

(f ⋅ g)(x) represents the product of f(x) and g(x), or the multiplication of their respective values at a specific x.

Step 2: Evaluating the equation

To find the area of the triangle when x = 2, we substitute x = 2 into the equation.

A(2) = 0.5 * (f ⋅ g)(2)

Now, let's substitute the functions f(x) and g(x) with their corresponding values at x = 2.

f(2) = 2² + 3 = 4 + 3 = 7

g(2) = 2(2) = 4

Substituting these values into the equation:

A(2) = 0.5 * (7 ⋅ 4)

= 0.5 * 28

= 14

Therefore, when x = 2, the area of the triangle is 14.

Step 3: Interpretation and Conclusion

The equation A(x) = 0.5 * (f ⋅ g)(x) correctly represents the area of the triangle. It takes into account the height function f(x) = x² + 3 and the base function g(x) = 2x, multiplied together and multiplied by 0.5 (or divided by 2), which is a common factor in the formula for the area of a triangle.

Hence, the equation A(x) = 0.5 * (f ⋅ g)(x) can be used to represent the area of the triangle for any given value of x.

Answer 2

B)
`A=0.5(f\cdot g)(x)`

Explanation:

`\displaystyle A=(1)/(2)bh\\\\A=(1)/(2)(2x)(x^2+3)\\\\A=(1)/(2)g(x)f(x)\\\\A=(1)/(2)(f\cdot g)(x)`