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Zina [86]
3 years ago
15

In Δ A B C, the measure of the largest angle is 32 less than 3 times the smallest angle. The measure of the middle angle is 8 mo

re than half the measure of the largest angle. What is the measure of the middle angle?
Mathematics
1 answer:
AlladinOne [14]3 years ago
5 0

Answer:

52degrees

Step-by-step explanation:

Let the measure of the largest angle be x

middle angle be y

smallest angle be z

Since the sum of angle in a triangle is 180 degrees, hence;

x + y + z = 180 .... 1

If the measure of the largest angle is 32 less than 3 times the smallest angle, then;

x = 3z - 32

3z = x+ 32

z = x+32/3 ..... 2

If he measure of the middle angle is 8 more than half the measure of the largest angle, then;

y = x/2 + 8 .... 3

Substitute 2 and 3 into 1;

x + x/2 + 8 + (x+32)/3 = 180

multiply through by 6

6x + 3x + 48 + 2(x+32) = 1080

9x + 48 + 2x + 64 = 1080

11x + 112 = 1080

11x = 1080-112

11x = 968

x = 968/11

x = 88 degrees

Get the middle angle y;

Since y = x/2 + 8

y = 88/2 + 8

y = 44 + 8

y = 52 degrees

Hence the measure of the middle angle is 52degrees

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Answer:

(a) 0.343

(b) 0.657

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(d) 0.216

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Step-by-step explanation:

Let P(a vehicle passing the test) = p

                        p = \frac{70}{100} = 0.7  

Let P(a vehicle not passing the test) = q

                         q = 1 - p

                         q = 1 - 0.7 = 0.3

(a) P(all of the next three vehicles inspected pass) = P(ppp)

                           = 0.7 × 0.7 × 0.7

                           = 0.343

(b) P(at least one of the next three inspected fails) = P(qpp or qqp or pqp or pqq or ppq or qpq or qqq)

      = (0.3 × 0.7 × 0.7) + (0.3 × 0.3 × 0.7) + (0.7 × 0.3 × 0.7) + (0.7 × 0.3 × 0.3) + (0.7 × 0.7 × 0.3) + (0.3 × 0.7 × 0.3) + (0.3 × 0.3 × 0.3)

      = 0.147 + 0.063 + 0.147 + 0.063 + 0.147 + 0.063 + 0.027

      = 0.657

(c) P(exactly one of the next three inspected passes) = P(pqq or qpq or qqp)

                 =  (0.7 × 0.3 × 0.3) + (0.3 × 0.7 × 0.3) + (0.3 × 0.3 × 0.7)

                 = 0.063 + 0.063 + 0.063

                 = 0.189

(d) P(at most one of the next three vehicles inspected passes) = P(pqq or qpq or qqp or qqq)

                 =  (0.7 × 0.3 × 0.3) + (0.3 × 0.7 × 0.3) + (0.3 × 0.3 × 0.7) + (0.3 × 0.3 × 0.3)

                 = 0.063 + 0.063 + 0.063 + 0.027

                 = 0.216

(e) Given that at least one of the next 3 vehicles passes inspection, what is the probability that all 3 pass (a conditional probability)?

P(at least one of the next three vehicles inspected passes) = P(ppp or ppq or pqp or qpp or pqq or qpq or qqp)

=  (0.7 × 0.7 × 0.7) + (0.7 × 0.7 × 0.3) + (0.7 × 0.3 × 0.7) + (0.3 × 0.7 × 0.7) + (0.7 × 0.3 × 0.3) + (0.3 × 0.7 × 0.3) + (0.3 × 0.3 × 0.7)

= 0.343 + 0.147 + 0.147 + 0.147 + 0.063 + 0.063 + 0.063

                  = 0.973  

With the condition that at least one of the next 3 vehicles passes inspection, the probability that all 3 pass is,

                         = \frac{P(all\ of\ the\ next\ three\ vehicles\ inspected\ pass)}{P(at\ least\ one\ of\ the\ next\ three\ vehicles\ inspected\ passes)}

                         = \frac{0.343}{0.973}

                         = 0.353

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