Lumber that is used to frame walls of houses is frequently sold in multiples of 2 ft. If the length of a board is not exactly a multiple of 2 ft, there is often no charge for the additional length. For example, if a board measures at least 8 ft, but less than 10 ft, then the consumer is charged for only 8 ft. &nbsp;(a) Suppose that the cost of lumber is $0.70 every 2 ft. For the function f(x) that determines the price of a board x feet long, find its formula.&nbsp;10≤x≤16.&nbsp;(b) Calculate the cost of boards with lengths of 8.6 ft and 15.1 ft.

Question

Lumber that is used to frame walls of houses is frequently sold in multiples of 2 ft. If the length of a board is not exactly a multiple of 2​ ft, there is often no charge for the additional length. For​ example, if a board measures at least 8​ ft, but less than 10​ ft, then the consumer is charged for only 8 ft. ​&amp;nbsp;(a) Suppose that the cost of lumber is ​$0.70 every 2 ft. For the function f(x) that determines the price of a board x feet long, find its formula.&amp;nbsp;10≤x≤16.&amp;nbsp;​(b) Calculate the cost of boards with lengths of 8.6 ft and 15.1 ft.

Clelioo · Accepted Answer

Step 1To express f(x) explicitly as an expression and use it for computing the required valuesStep 2Consider the standard x case, where x is larger than or equal to 10 but strictly less than 12. The cost is the same for all x in this range&amp;nbsp;=5&amp;nbsp;times 0.70. This can be captured by the formula (using the floor function)First consider10≤x&amp;lt;12f(x)=cos⁡tf&amp;nbsp;or&amp;nbsp;10ft=5×0.70=[x2]×0.70,where&amp;nbsp;[x2]=&amp;nbsp;floor&amp;nbsp;(x2)&amp;nbsp;integral part&amp;nbsp;(x2)Step 3The formula is valid for all positive integers x greater than or equal to 2, especially within the specified range.10≤x≤16f(x)=[x2]×0.70,where&amp;nbsp;[x2]=&amp;nbsp;floor&amp;nbsp;(x2)=&amp;nbsp;integral part&amp;nbsp;(x2)Step 4More specifically, the formula is calculated for the specified range.f(x)=[x2]×0.70,=$3.50,&amp;nbsp;if&amp;nbsp;10≤x&amp;lt;12=$4.20,&amp;nbsp;if&amp;nbsp;12≤x&amp;lt;14=$4.90,&amp;nbsp;if&amp;nbsp;14≤x&amp;lt;16=$5.60,&amp;nbsp;if&amp;nbsp;x=16Step 5Result: (b)b)&amp;nbsp;f(8.6)=f(8)=4×0.70=$2.80f(15.1)=f(14)=7×0.70=$4.90

Lumber that is used to frame walls of houses is frequently sold in mul

Answered question

Answer & Explanation

New Questions in Pre-Algebra