Question

a fish has a triangular tooth with a height that is 6 centimeters longer than the base. if the area of the tooth is 8 square centimeters, find its base and height. the base of the tooth is 2 centimeters. (type an integer or a fraction.) the height of the tooth is (type an integer or a fraction.)

Explanation:

Step1: Recall area formula

The area formula for a triangle is $A=\frac{1}{2}bh$. Here, let the base $b = x$ and the height $h=x + 6$, and $A = 8$. So we have the equation $\frac{1}{2}x(x + 6)=8$.

Step2: Simplify the equation

Multiply both sides of the equation by 2 to get $x(x + 6)=16$. Expand the left - hand side: $x^{2}+6x=16$. Rearrange it to the standard quadratic form $x^{2}+6x - 16=0$.

Step3: Solve the quadratic equation

Factor the quadratic equation $x^{2}+6x - 16=(x + 8)(x - 2)=0$. Then $x+8 = 0$ or $x - 2=0$. Solving for $x$, we get $x=-8$ or $x = 2$. Since the length cannot be negative, we take $x = 2$.

Step4: Find the height

The height is $x + 6$. Substitute $x = 2$ into it, we get $h=2 + 6=8$.

Answer:

8