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iragen [17]
4 years ago
9

Reyna's house is 12 blocks due west of the school. Shaquille's house is 5 blocks due north of the school. Shaquille walks to Rey

na's house every morning and then they walk to school together. What is the shortest distance Shaquille could walk on his way to school when he walks with Reyna?
Mathematics
1 answer:
suter [353]4 years ago
5 0

Answer:

The shortest distance that Shaquille could walk on his way to school when he walks with Reyna is = 25 blocks.

Step-by-step explanation:

Reyna's house is 12 blocks due west of the school. Shaquille's house is 5 blocks due north of the school.

So, Reyna's house, the school, and the Shaquille's house form a right triangle and the shortest distance from Shaquille's house to Reyna's house is the hypotenuse of the right triangle.

So, the shortest distance that Shaquille could walk on his way to school when he walks with Reyna is = \sqrt{12^{2} + 5^{2}} + 12 = 13 + 12 = 25 blocks. (Answer)

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Problems 2.21, 2.22, 2.23
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Statements can be proved by contrapositive, contradiction or by induction.

  • <em>2.21 and 2.23 are proved by contrapositive</em>
  • <em>2.22 is proved by induction</em>

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<u>2.21: If </u>n^3<u> is even, then n is even (By contrapositive)</u>

The contrapositive of the above statement is that:

<em>If n is odd, then  </em>n^3<em> is odd</em>

Represent the value of n as:

n = 2k + 1, where k \ge 0

Take the cube of both sides

n^3 = (2k + 1)^3

Expand

n^3 = 8k^3 + 6k^2 + 6k + 1

Group

n^3 =[ 8k^3 + 6k^2 + 6k] + 1

Factor out 2

n^3 =2[4k^3 + 3k^2 + 3k] + 1

Assume w is an integer; where:

w =4k^3 + 3k^2 + 3k

So, we have:

n^3 =2w + 1

The constant term (i.e. 1) means that n^3 is odd.

Hence, the statement has been proved by contrapositive.

<em>i.e. If n is odd, then  </em>n^3<em> is odd</em>

<u />

<u>2.22  </u>3n + 4<u> is even, if and only if n is even</u>

We have: 3n + 4<u />

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Assume that: n = 2k + 2 for k \ge 0

So, we have:

3n + 4 = 3(2k + 2) + 4

Open bracket

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Factorize

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The factor of 2 means that 3n + 4 is even.

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<u>2.22: </u>s \ne -1<u> and </u>t \ne -1<u>, then </u>s + t + st \ne -1<u />

To do this, we prove by contrapositive.

The contrapositive of the above statement is:

If s = -1 and t=-1, then s + t + st = -1

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Substitute the values of s and t in: s + t + st = -1

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Hence, by contrapositive:

If s = -1 and t=-1, then s + t + st = -1

Read more about proofs  at:

brainly.com/question/19643658

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