Question

The height of a triangle is 4 inches greater than twice its base. The area of the triangle is no more than 168 in.² which inequality can be used to find the possible lengths, x, of the base of the triangle?
A. x(x+2)=>168
B. x(x+2)=<168
C. 1/2x(x+4)=<168
D. 1/2x(x+4)=>168

Answer 1

Answer:

Correct option: B

Step-by-step explanation:

When we say a is no more than b, we express this in a mathematical language as follows:

$a\leq b$

In this inequality, we know that the area of the triangle is no more than 168 in². In other words, if the area is named $A$, then:

$\mathbf{(1)} \ A\leq 168$

We also know that the height of a triangle is 4 inches greater than twice its base. Translating this in a mathematical language:

$\mathbf{(2)} \ h=2b+4 \\ \\ h:height \ of \ the \ triangle \\ \\ b:base \ of \ the \ triangle$

From geometry, we know that the area of a triangle is given by:

$\mathbf{(3)} \ A=\frac{bh}{2}$

Matching (1), (2) and (3):

$\frac{bh}{2}\leq 168$

Since the length of the base of the triangle is $x$, then $b=x$

$\frac{x(2x+4)}{2}\leq 168 \\ \\ Common \ factor \ 2: \\ \\ \frac{2x(x+2)}{2}\leq 168 \\ \\ \boxed{x(x+2)\leq 168}$

Finally, correct option is B.