Practice Set 19 Problem 1 Determine that depth of the machine groove in this steel block. The groove is really an isosceles triangle with two equal sides. We're going to draw a line across the top to fill in the gap, then determine its length. The entire span is 55 millimeters. We know we have a 20 so we'll subtract 20 here.
We'll subtract the right hand side of that gap. Twenty-three from the total. That'll give us that gap span and that results and that results in a length of 12 millimeters.
So our gap is 12 millimeters. Next we will draw an altitude from the vertex to the line we just drew. This altitude bi-sects that angle of the groove and it also bi-sects the gap or the line we drew and resulting in 2 equal triangles. Let me draw one of those triangles in large. So we can see the detail of what just happened.
I got this right hand side of this triangle here. This altitude bi-sected this gap length 12 which means we have six from one side of that point of the altitude and six on the other side. So translating that we have a length of our right triangle as six here.
The altitude drawn in an isosceles triangle bi-sects the angle so the original groove angle was 40 degrees. Half of that would mean we have an acute angle of twenty degrees and our challenge is to determine the depth which would be length of this altitude that we drew. Solving this right triangle we will use our acute angle that we know the measurement of for our reference angle.
Starting with labeling the sides then hypotenuse opposite the right angle A our reference angle thru the triangle this side is the opposite leaving the side we're looking for as the adjacent side. Since we're working with opposite an adjacent sides that means we will need the tangent formula. So tangent of the angle is equal to the opposite over the adjacent. Plugging in the values that we have tangent of 20 degrees is equal to the opposite 6 over the adjacent our unknown. We'll multiply both sides by d to get the variable out of the denominator and eliminate the fraction. Next we want to get the variable by itself.
It's being multiplied by the tangent of twenty. So to undo multiplication we will divide both sides by the tangent of twenty degrees. Twenty degrees the tangent of and tangent of twenty degrees are the same thing. They cancel one another out leaving d and we will go to the calculator then to simplify the value that we have here and do the indicated division. I have a direct algebraic logic calculator downloaded on an iPad Air and I will start with the numerator 6 dividing it by the tangent of twenty. So I will tap tangent and since this is a DAL calculator we do the trig functions first and then the angle measurement. You see I don't have an answer so I have to hit the equal so that that division is calculated. This gives us a value for D the depth of our cut in the problem rounding to the nearest tenth of 16.5 millimeters.
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