SOLUTION: BUILDING A SKI RAMP. WRITE AN EQUALITY IN THE VARIABLE X FOR THE DEGREE MEASURE OF THE SMALLEST ANGLE OF THE TRIANGLE, GIVEN THE DEGREE MEASURE OF THE SMALLEST ANGLE IS AT MOST 30

Question 175945: BUILDING A SKI RAMP. WRITE AN EQUALITY IN THE VARIABLE X FOR THE DEGREE MEASURE OF THE SMALLEST ANGLE OF THE TRIANGLE, GIVEN THE DEGREE MEASURE OF THE SMALLEST ANGLE IS AT MOST 30 DEGREE. First ANGLE =X, Second ANGLE = X+8
Answer by gonzo(654) (Show Source): You can put this solution on YOUR website!
let x be the smallest angle.
let x + 8 be the second angle.
let x + y = the third angle.
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since the sum of the angles of a triangle equal 180 degrees.
this simplifies to
this simplifies to
where:
x>0
x<=30
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if x >0, then:
(180-y-8)/2 > 0
multiply both sides of this equation by 2 to get:
180 - y - 8 > 0
add y to both sides of the equation and combine like terms to get:
172 > y
subtract y from both sides of this equation and subtract 172 from both sides of this equation to get:
-y > - 172
multiply both sides of this equation by -1 to get:
y < 172
please note that multiplying both sides of an inequality by a minus number changes the inequality.
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so we know that y has to be smaller than 172.
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if x <= 30, then:
(180-y-8)/2 <= 30
multiply both sides by 2 to get:
180 - y - 8 <= 60
add y to both sides and subtract 60 from both sides to get:
180 - 60 - 8 <= y
combine like terms to get:
112 <= y
subtract 112 from both sides of the equation and subtract y from both sides of the equation to get:
-y <= -112
multiply both sides of the equation by -1 to get:
y >= 112
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looks like x will be in the range of:
0 < x <= 30
when y is in the range of:
112 <= y < 172
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to prove this is true, take several values of y in and out of the acceptable range and see what happens to x.
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when y = 172, x = 0.
since y is outside of the acceptable range, x is also outside of the acceptable range so this is good.
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when y = 112, x = 30.
since y is within the acceptable range, x is also within the acceptable range so this is good.
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when y = 150, x = 11.
since y is within the acceptable range, x is also within the acceptable range so this is good.
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when y = 200, x = -14.
since y is outside of the acceptable range, x is also outside of the acceptable range so this is good.
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when y = 110, x = 31.
since y is outside of the acceptable range, x is also outside of the acceptable range so this is good.
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the limits for y lead to the proper limits for x so the answer is good.
the equation to solve this problem is:

the requirements for x are:
0 < x <= 30
the requirements for y in order to satisfy the requirements for x are:
112 <= y < 172
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as long as y is in the acceptable range, x will be in the acceptable range.
as long as y is outside of the acceptable range, x will be outside of the acceptable range.
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since they asked for the equality equation for the smallest angle, your answer has to be:

with the stipulation that:
112 <= y < 172
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i don't think you can solve this problem without introducing another variable since the range of that variable is important to the solution.
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if you had stipulated what the third angle had to be in terms of x, that would have been different.
then we would have had a value for y and we could have used that value in place of y.
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