To solve for the height of the triangular shelf given its area and base, we can use the formula for the area of a triangle. The formula states that the area [tex]\( A \)[/tex] of a triangle is given by:
[tex]\[ A = \frac{1}{2} \times \text{base} \times \text{height} \][/tex]
In this problem, we are given the area [tex]\( A = 105 \, \text{in}^2 \)[/tex] and the base [tex]\( \text{base} = 14 \, \text{in} \)[/tex]. We need to find the height [tex]\( h \)[/tex] of the triangle.
Substituting the given values into the formula, we get:
[tex]\[ 105 = \frac{1}{2} \times 14 \times h \][/tex]
Now, we solve for [tex]\( h \)[/tex] step-by-step:
1. Simplify the right side of the equation:
[tex]\[ 105 = 7h \][/tex]
This simplification comes from multiplying [tex]\( \frac{1}{2} \times 14 \)[/tex], which equals 7.
2. To isolate [tex]\( h \)[/tex], divide both sides of the equation by 7:
[tex]\[ h = \frac{105}{7} \][/tex]
3. Perform the division:
[tex]\[ h = 15 \][/tex]
Therefore, the height of the triangular shelf is [tex]\( 15 \, \text{in} \)[/tex]. The correct answer is:
[tex]\[ \boxed{15 \, \text{in}} \][/tex]
