What formula could I use to calculate the apparent weight of an object on an angled surface, e.g, a 100 kg box on the side of a hill at 10° given that the measuring instrument is also at that same 10° from level?
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$\begingroup$ You need to think first that force is a vector. On a plane surface you have the weight as you read it in the scale. If you have it on a tilted surface, then the weight is "distributed" differently along different directions. $\endgroup$– user2820579Apr 24, 2019 at 17:10
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2 Answers
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Note :- reading of a instrument like weighing machine is not the actual weight of the object placed on it but the normal force applied but the object on the scale .
so now from the image we can see for equilibrium of the block normal to incline we have $ R = mg cos( \theta) $ hence the reding would be not equal to the weight of the object but a little less than it. 
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The apparent weight is just the normal force. Using geometry you can figure out that the normal force is given by $$F = mg\cos\theta.$$