The height of a triangle is 4 inches more than twice the length of the base. The area of the triangle is 35 square inches. What is the height of the triangle?
$14$Let the Base of the $triangle$ be $color(red)(x$
Then the Height will be $color(red)(2x+4$
Area of $triangle$=$color(brown)(1/2bh$
Where,
$color(brown)(b=base,h=height,Area=35$ (in this case)
Substitute the values into the equation
$rarr35=1/2(2x+4)(x)$
$rarr35=((cancel2x+cancel4))/cancel2(x)$
$rarr35=(x+2)(x)$
$rarr35=x^2+2x$
Subtract $35$ both sides
$rarr0=x^2+2x-35$
Rewrite the equation in the Standard form
$x^2+2x-35=0$
Factor the equation
$rarr(x+7)(x-5)=0$
So we have $color(blue)(x=-7,5$
length or distance should not be $uln$$uleulgulaultuliulvule$ numbers
So $color(orange)(x=5$
They have asked us to find the Height
So,
$rArrcolor(green)(Height=2x+4=2(5)+4=10+4=14$height $= 14$ inches.Let the height be $h$ and the base be $h$ (inches)
$h=2b+4$
Area: $(bh)/2=35$
$color(white)(XXX)bxx(2b+4)=70$
$color(white)(XXX)2b^2+4b=70$
$color(white)(XXX)b^2+2b-35=0$
$color(white)(XXX)(b-5)(b+7)=0$
$rArr b=5 or b=-7$
Since the base must be positive:
$color(white)(XXX)b=5$
and
$color(white)(XXX)h=2b+4=14$
