Select the correct answer. The length, [tex]l[/tex], of the shadow cast by an object varies directly as the height, [tex]h[/tex], of the object. If [tex]k[/tex] is the constant of variation, which equation represents this situation?
A. [tex]1+h=k[/tex]
B. [tex]l h=k[/tex]
C. [tex]l=h k[/tex]
D. [tex]l=h+k[/tex]
Answer (2)
The problem describes a direct variation between the length of a shadow ( l ) and the height of an object ( h ).
Direct variation is expressed as l = k × h , where k is the constant of variation.
Comparing this with the given options, the correct equation is l = hk .
Therefore, the answer is C .
Explanation
Understanding the Problem The problem states that the length, l , of the shadow cast by an object varies directly as the height, h , of the object. This means that l is proportional to h . The constant of variation is given as k . We need to find the equation that represents this direct variation.
Expressing Direct Variation When two quantities vary directly, it means that one quantity is a constant multiple of the other. In this case, since l varies directly as h , we can write the relationship as: l = k × h where k is the constant of variation.
Comparing with Options Now, let's compare this equation with the given options:
A. l + h = k B. l h = k C. l = hk D. l = h + k
Option C, l = hk , matches the equation we derived, l = k × h . Note that hk is the same as k × h .
Conclusion Therefore, the correct equation that represents the direct variation between the length of the shadow and the height of the object is l = hk .
Examples
Direct variation is a fundamental concept in many real-world scenarios. For instance, the amount you earn at a job with a fixed hourly rate varies directly with the number of hours you work. If you earn 15 p er h o u r , yo u r t o t a l e a r nin g s ( E ) c anb ere p rese n t e d a s E = 15h , w h ere h i s t h e n u mb ero f h o u rs w or k e d . S imi l a r l y , t h e d i s t an ce a c a r t r a v e l s a t a co n s t an t s p ee d v a r i es d i rec tl y w i t h t im e . I f a c a r t r a v e l s a t 60 mi l es p er h o u r , t h e d i s t an ce ( D ) i t co v ersc anb ere p rese n t e d a s D = 60t , w h ere t$ is the time in hours. Understanding direct variation helps in predicting outcomes based on proportional relationships.
The equation that represents the direct variation between the length of the shadow and the height of the object is l = hk , which corresponds to option C. This describes how the shadow's length increases proportionally with the height of the object. Therefore, the selected answer is option C.
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