Chapter 1.8, Problem 44PPS

(a)

To determine

To write: An algebraic expression to represent the number of times shadow is shown in $y$ years.

Answer to Problem 44PPS

Thealgebraic expression is $0.85y$ .

Explanation of Solution

Given information:

It is given that the most famous groundhog, Punxsutawney Phil in Pennsylvania sees his shadow $85\%$ of the time.

Calculation:

The most famous groundhog, Punxsutawney Phil in Pennsylvania, sees his shadow $85\%$ of the time.

The algebraic expression representing the number of times he sees his shadow in $y$ years is $0.85y$ .

Therefore, the algebraic expression is $0.85y$ .

(b)

To determine

To check: The given scenario is true of false for the given conditional statement.

Answer to Problem 44PPS

Thetrue or false is shown in table (1).

Explanation of Solution

Given information:

The statement are“if Groundhog sees its shadow, then there will be 6 more weeks of winter” and “if it does not see its shadow, then there will be an early spring.”

Calculation:

On Groundhog day, if Groundhog sees his shadow, then there will be 6 more weeks of winter. If it does not see his shadow, then there will be an early spring.

According to the given statement the following table is completed.

Sees his shadow or not	6 more weeks of winter or an early spring	True or false
Shadow	Winter	True
Shadow	Spring	False
No Shadow	Winter	False
No Shadow	Spring	True

Table (1)

Hence, the true or false is shown in table (1).

(c)

To determine

The situations listed in the table could be considered as a counterexample to the original statement.

Answer to Problem 44PPS

Thesecond and third situations explain the counterexample for the given statements.

Explanation of Solution

Given information:

The statement are “if Groundhog sees its shadow, then there will be 6 more weeks of winter” and “if it does not see its shadow, then there will be an early spring.”

Calculation:

Consider the first statement.

“If Groundhog sees its shadow, then there will be 6 more weeks of winter.”

One counter example to this statement from the table is as follows.

“If Groundhog sees his shadow, then there will be an early spring” according to second row of table (1).

Consider the second statement.

“If it does not see its shadow, then there will be an early spring.”

One counter example to this statement from the table is as follows.

“If Groundhog sees its shadow, then there will be 6 more weeks of winter” according to third row of table (1).