Answer:

base = 12 ft
height = 29 ft

Step-by-step explanation:

Let:

b = base of the triangular boat sail

h = height of the triangular boat sail

SOLUTION:

Since the height is 5 ft more than twice the base, we can represent the height as:

h = 2b + 5

The area of the sail boat is:

A = 174 ft²

The formula for the area of a triangle is:

[tex]\mathsf{A=\dfrac{1}{2}bh}[/tex]

Using the formula for the area and substituting the given values we get:

[tex]\mathsf{A=\dfrac{1}{2}bh}[/tex]

[tex]\mathsf{174 = \dfrac{1}{2}bh}[/tex]

substitute h = 2b + 5

[tex]\mathsf{174=\dfrac{1}{2}b(2b+5)}[/tex]

simplify by multiplying both sides by 2

[tex]\mathsf{2(174)=2\left[\dfrac{1}{2}b(2b+5)\right]}[/tex]

[tex]\mathsf{348=b(2b+5)}[/tex]

distribute b

[tex]\mathsf{348=2b^2+5b}[/tex]

transpose 348 to the right hand side

[tex]\mathsf{0=2b^2+5b-348}[/tex]

rearrange

[tex]\mathsf{2b^2+5b-348=0}[/tex] (QUADRATIC EQUATION)

Solving for b by factoring, we get:

[tex]\mathsf{(2b+29)(b-12)=0}[/tex]

[tex]\mathsf{b_1=-\dfrac{29}{2}}[/tex]

[tex]\mathsf{b_2=12}[/tex]

Disregard the negative value since the base of the sail boat cannot be negative.

b = 12 ft (ANSWER)

Solving for h using the relationship between the height and the base

h = 2b + 5

h = 2(12) + 5

h = 29 ft (ANSWER)

CHECKING:

[tex]\mathsf{A=\dfrac{1}{2}bh}[/tex]

[tex]\mathsf{174=\dfrac{1}{2}(12)(29)}[/tex]

174 = 174 (OK)