The base of the triangle is one and a half times it's height. what is height if area is 75

 The height is: 10.61 units. 
                       {or, write as:  "(7.5 √2)  units"}. 

Explanation:

Area of a triangle:  A = 1/2 * base * height = 1/2 * b * h ;

Given:  b = 1.5 * h ;
           A = 75 units²

Solve for "h" ("height") ;  h = 1.5 b ;

So, we solve for "b" ; then we plug that value into the equation: 
  
        →   h = 1.5 b ;  to get the height, "h" ;

        →   75 = 1/2 * (b) (1.5 b) ;

         Multiply the ENTIRE equation by "2" ;
  to get rid of the fraction and decimals ; 

       →   2 * { 75 = 1/2 * (b) (1.5 b) } ;

       →  150  =  b * 3b ;

       →  150  =  3 b²  ;  
 
       ↔ 3 b²  =  150 ; 

 
            Now, divide EACH side of the equation by "3" ;

       →  3 b² / 3 = 150 / 3 ; 

       →     b²  =  50 ;

           Now, take the POSITIVE square root of each side of the equation;
to isolate "b" on one side of the equation; and to solve for "b" ;

        →   √(b²)  =  √50

        →      b  =  √50

        →     √50 = √25 *√2 = 5√2

        h = (1.5) b = (1.5) *(5) (

Explanation:

Area of a triangle:  A = 1/2 * base * height = 1/2 * b * h ;

Given:  b = 1.5 h ;
           A = 75 units²

Solve for "h" ("height") ;  h = 1.5 b ;

So, we solve for "b" ; then we plug that value into the equation: 
  
        →   h = 1.5 b ;  to get the height, "h" ;

        →   75 = 1/2 * (b) (1.5 b) ;

         Multiply the ENTIRE equation by "2" ; 
  to get rid of the fraction and decimals ; 

       →   2 * { 75 = 1/2 * (b) (1.5 b) } ;

       →  150  =  b * 3b ;

       →  150  =  3 b²  ;  
 
       ↔ 3 b²  =  150 ; 

 
            Now, divide EACH side of the equation by "3" ;

       →  3 b² / 3 = 150 / 3 ; 

       →     b²  =  50 ;

           Now, take the POSITIVE square root of each side of the equation; 
to isolate "b" on one side of the equation; and to solve for "b" ;

        →   √(b²)  =  √50

        →      b  =  √50

        →      b  =  √50 = √25 *√2 = 5√2

        →      h = 1.5 * 5 * √2 ;

       →    h = (7.5 √2)  units ;
                           
                      or,  (7.5) * (√2) = 10.6066017177982129 units ;

                                            → round to 10.61 units .

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