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Sagot :
Given:
Base of triangle = b
Height of triangle, h, is 3 feet less than twice its base. This is expressed as:
h = 2b - 3
Area of triangle = 52 ft²
To find the height of the triangle, use the Area of a triangle formula below:
[tex]A=\frac{1}{2}bh[/tex]Thus, we have:
[tex]\begin{gathered} 52=\frac{1}{2}\times b\times(2b-3) \\ \\ 52=\frac{b(2b-3)}{2} \end{gathered}[/tex]Let's solve for the base, b:
[tex]\begin{gathered} 52=\frac{2b^2-3b}{2} \\ \\ Multiply\text{ both sides by 2:} \\ 52\times2=\frac{2b^2-3b}{2}\times2 \\ \\ 104=2b^2-3b \end{gathered}[/tex]Subtract 104 from both sides to equate to zero:
[tex]\begin{gathered} 2b^2-3b-104=104-104 \\ \\ 2b^2-3b-104=0 \end{gathered}[/tex]Factor the quadratic equation:
[tex](2b+13)(b-8)[/tex]Thus, we have:
[tex]\begin{gathered} (2b+13)\text{ = 0} \\ 2b\text{ + 13 = 0} \\ 2b=-13 \\ b=-\frac{13}{2} \\ \\ \\ (b-8)=0 \\ b=8 \end{gathered}[/tex]We have the possible values for b as:
b = - 13/2 and 8
Since the base can't be a negative value, let's take the positive value.
Therefore, the base of the triangle, b = 8 feet
To find the height, substitute b for 8 from the height equation, h=2b-3
Thus,
h = 2b - 3
h = 2(8) - 3
h = 16 - 3
h = 13 feet.
Therefore, the height of the triangle, h = 13 feet
ANSWER:
13 feet
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