Question

a triangles height is 6c meters and the base is c - 1 meters. brian knows that the area of the triangle is 18 square meters and is trying to find the height. which shows the equation brian can use and the correct measure of the height of the triangle?
area of a triangle = 0.5(base)(height)
0.5(c - 1)(6c) = 18; height = 3 meters
0.5(c - 1)(6c) = 18; height = 18 meters
(c - 1)(6c) = 18; height = 3 meters
(c - 1)(6c) = 18; height = 18 meters

Explanation:

Step1: Substitute into area formula

The area formula of a triangle is \(A = 0.5\times(\text{base})\times(\text{height})\). Given \(A = 18\), base \(=c - 1\), height \(=6c\). Substituting these values, we get \(0.5(c - 1)(6c)=18\).

Step2: Solve the equation for \(c\)

First, simplify the left - hand side of the equation \(0.5(c - 1)(6c)=18\).
\(0.5\times6c(c - 1)=18\), \(3c(c - 1)=18\), \(3c^{2}-3c=18\), divide both sides by \(3\): \(c^{2}-c = 6\), \(c^{2}-c-6=0\).
Factor the quadratic equation: \((c - 3)(c + 2)=0\).
Set each factor equal to zero: \(c-3 = 0\) or \(c + 2=0\).
Solving \(c-3 = 0\) gives \(c = 3\), and solving \(c + 2=0\) gives \(c=-2\). Since \(c\) represents a length - related quantity (because base \(=c - 1\) and height \(=6c\)), \(c>0\), so \(c = 3\).

Step3: Calculate the height

Substitute \(c = 3\) into the height formula \(h = 6c\). Then \(h=6\times3=18\) meters.

Answer:

B. \(0.5(c - 1)(6c)=18\); height \( = 18\) meters