To determine the correct equation representing the given direct variation scenario, we need to understand the concept of direct variation. The statement mentions that the length [tex]\( l \)[/tex] of the shadow cast by an object varies directly as the height [tex]\( h \)[/tex] of the object. This means that [tex]\( l \)[/tex] and [tex]\( h \)[/tex] are directly proportional to each other, and can be represented by the equation:
[tex]\[ l = kh \][/tex]
where [tex]\( k \)[/tex] is the constant of variation.
Let's evaluate the given options:
A. [tex]\( 1 + h = k \)[/tex]: This equation does not represent direct variation, as it does not involve multiplication between the variables [tex]\( l \)[/tex] and [tex]\( h \)[/tex].
B. [tex]\( t h = k \)[/tex]: The variable [tex]\( t \)[/tex] is not defined in the context of the problem, making this option irrelevant to the scenario.
C. [tex]\( I = h k \)[/tex]: This equation represents a form similar to direct variation. However, the variable [tex]\( I \)[/tex] should be [tex]\( l \)[/tex] to correctly match the problem statement. Interpreting [tex]\( I \)[/tex] as [tex]\( l \)[/tex], this seems the closest, though there is a typographical error.
D. [tex]\( l = h + k \)[/tex]: This equation represents a linear relationship but not direct variation, as direct variation should involve multiplication, not addition.
The correct equation that matches the scenario described is:
[tex]\[ l = kh \][/tex]
Given the options and considering the most appropriate representation that fits [tex]\( l = kh \)[/tex], the correct answer corresponds to option C (although with a minor adjustment, interpreting [tex]\( I \)[/tex] as [tex]\( l \)[/tex]).
Therefore, the correct answer is:
[tex]\[ \boxed{3} \][/tex]
