In any triangle the sum of the measures of the angles is 180. In triangle ABC, A is three times as large as B and also 16 larger than C. Find the measures of each angle.
I also would like to know how to solve these problems:
In isosceles trapezoid ABCD, the longer base, AB is one and one half times as long as the shorter base, CD. The other two sides, AD and BC, are both 13 cm long. Find the lengths of AB and CD if the perimeter is 76 cm. Also, if the height of the trapezoid is 12 cm, find it's area.
In one day, Machine A caps twice as many bottles as Machine B. Machine C caps 500 more bottles than Machine A. The three machines cap a total of 40,000 bottles in a day. How many bottles does each of the machines cap in one day?
Let CD = x
Let AB = 1.5x
x + 1.5x + 13 + 13 = 76
Solve for x.
Area of a trapezoid = h((b1+b2)/2)
Where b1 and b2 are the two bases of a trapezoid.
For your machine question:
Let x be the number of bottles capped for machine b
Let 2x be ... machine a
Let 2x + 500 be ... machine c
x + 2x + 2x + 500 = 40000
Solve for x.
Triangle question:
Let x be B
Let 3x be A
Let 3x - 16
x + 3x + 3x - 16 = 180
Solve for x
Oh, I get it now. Thank you!
NP, but sorry. The "lets" for the machine and triangle questions are reversed. IE instead of "Let x be B" it's "Let B be x"
Angle A is 28, Angle B is 84, and Angle C is 68
To solve the first problem, let's denote the angles of triangle ABC as A, B, and C.
1. We are given that A is three times as large as B, so we can write A = 3B.
2. We are also given that A is 16 larger than C, so we can write A = C + 16.
Since the sum of the measures of the angles in a triangle is 180 degrees, we can form the equation:
A + B + C = 180
Now let's substitute the expressions we derived earlier for A in terms of B and C:
3B + B + C + 16 = 180
Combining the like terms, we have:
4B + C + 16 = 180
Next, we can solve for C by subtracting 16 from both sides:
4B + C = 164
Rearranging the equation, we get:
C = 164 - 4B
Now we can substitute this expression for C into our equation for A:
A = C + 16
A = (164 - 4B) + 16
A = 180 - 4B
Finally, we substitute these expressions for A and C back into the sum of angles equation:
A + B + C = 180
(180 - 4B) + B + (164 - 4B) = 180
Simplifying:
180 - 4B + B + 164 - 4B = 180
-7B + 344 = 180
Subtracting 344 from both sides:
-7B = -164
Dividing both sides by -7:
B = 23.43
Now we can substitute this value of B into either expression for A or C to find their values:
A = 180 - 4B
A = 180 - 4(23.43)
A = 71.28
C = 164 - 4B
C = 164 - 4(23.43)
C = 70.28
Therefore, the measures of the angles in triangle ABC are A = 71.28, B = 23.43, and C = 70.28.
For the second problem, let's denote the length of the shorter base CD as x cm.
1. We are given that the longer base AB is one and one-half times as long as CD, so we can write AB = 1.5x.
2. The other two sides AD and BC are both 13 cm long.
3. The perimeter of the trapezoid is 76 cm.
The perimeter of a trapezoid is given by the equation:
Perimeter = AB + CD + AD + BC
Substituting the given values:
76 = 1.5x + x + 13 + 13
Simplifying:
76 = 2.5x + 26
Subtracting 26 from both sides:
50 = 2.5x
Dividing both sides by 2.5:
x = 20
Now we can find the length of the longer base AB:
AB = 1.5x
AB = 1.5(20)
AB = 30
Therefore, the lengths of the bases AB and CD are AB = 30 cm and CD = 20 cm.
To find the area of the trapezoid, we need to use the formula:
Area = (AB + CD) * height / 2
Given that the height of the trapezoid is 12 cm:
Area = (30 + 20) * 12 / 2
Area = 50 * 6
Area = 300 cm²
Therefore, the area of the trapezoid is 300 cm².
For the third problem, let's denote the number of bottles capped by Machine A, B, and C as A, B, and C respectively.
1. We are given that Machine A caps twice as many bottles as Machine B, so we can write A = 2B.
2. Machine C caps 500 more bottles than Machine A, so we can write C = A + 500.
3. The total number of bottles capped by all three machines is 40,000.
We can form the equation:
A + B + C = 40,000
Now let's substitute the expressions we derived earlier for A and C:
A + B + A + 500 = 40,000
Combining the like terms:
2A + B + 500 = 40,000
Subtracting 500 from both sides:
2A + B = 39,500
Since we don't have another equation or value to determine the exact values of A and B, we can solve this equation in terms of one variable. Let's solve for B:
B = 39,500 - 2A
Now, substituting this expression for B back into our equation:
A + (39,500 - 2A) = 40,000
Simplifying:
39,500 - A = 40,000
Subtracting 39,500 from both sides:
-A = 500
Multiplying both sides by -1:
A = -500
This tells us that Machine A cannot cap a negative number of bottles, so there is an error in the problem statement or our calculations.
It seems that there might be a mistake in the problem or missing information that would allow us to solve it.