Customer: “I would like to purchase 20 products at a cost of $65.00 per product. What will be my total?” Employee: “Your state h

Answer:

The amount of wool required for a sweater was 400 grams.

A hat needed 80 grams of wool.

A scarf required 75 grams of wool.

Step-by-step explanation:

Let

x ------> grams of wool necessary for a sweater

y -----> grams of wool necessary for a hat

z -----> grams of wool necessary for a scarf

From the information given

x+y+z=555 -----> equation A

y=x/5 ----> x=5y -----> equation B

y=z+5 ---> z=y-5 ----> equation C

By substituting equations B and C into equation A, we can solve for y:

(5y)+y+(y-5)=555

7y=560

y=80 grams

Now, find x

x=5(80)=400 grams

Next, determine z

z=80-5=75 grams

In conclusion,

400 grams of wool was required for the sweater,

the hat needed 80 grams of wool, and

the scarf required 75 grams of wool.

Response:

This function is continuous from the left at x=1 and from the right at x=0.

Detailed Explanation:

A function is said to be continuous from the right at a point if:

when x→a⁺ lim f(x)=f(a)

And continuous from the left if:

when x→a⁻ lim f(x)=f(a)

Thus, as the functions presented are continuous, we must search for points of discontinuity only when the functions transition.

At x=0:

when x→0⁺ lim f(x)=lim e^x = e^0 = 1

when x→0⁻ lim f(x)=lim (x+4) = (0+4) = 4

Therefore, since f(0) = e^0=1, the function is continuous from the right at x=0.

At x=1:

when x→1⁺ lim f(x)=lim (8-x) = (8-0) = 8

when x→1⁻ lim f(x)=lim e^x = e^1 = e

Hence, since f(1) = e^1=e, the function is continuous from the left at x=1.