Let's go through the problems in the image step by step.

Problem 1:

Problem: Kevin is designing a label for a can. The height of the label is 5 cm. If the area of the label is $120$ square centimeters, how long is the label?

Solution: The label is essentially a rectangle that wraps around the cylindrical can. The area $A$ of the rectangle can be found using the formula: $A = \text{Length} \times \text{Height}$ Given:

• Area $A = 120 \, \text{cm}^2$
• Height $h = 5 \, \text{cm}$

Let the length of the label be $L$.

Then, $120 = L \times 5$ To find $L$: $L = \frac{120}{5} = 24 \, \text{cm}$

So, the length of the label is 24 cm.

Problem 2:

Problem: A parallelogram has a height of $x + 3$ and an area of $25x + 75$. What is the measure of the base?

Solution: The area $A$ of a parallelogram is given by: $A = \text{Base} \times \text{Height}$ Given:

• Height $h = x + 3$
• Area $A = 25x + 75$

Let the base be $b$.

So, $25x + 75 = b \times (x + 3)$ To find $b$, divide both sides by $(x + 3)$: $b = \frac{25x + 75}{x + 3}$ Factor the numerator: $b = \frac{25(x + 3)}{x + 3}$ Cancel out $(x + 3)$ from the numerator and denominator: $b = 25$

So, the measure of the base is 25 units.

Would you like further details or have any questions? Here are five related questions:

1. How do you determine the perimeter of the label from Problem 1?
2. What happens to the base if the height in Problem 2 changes?
3. Can you generalize the formula for the area of a cylinder?
4. How do you handle cases where the label does not fully wrap around the cylinder?
5. If the area of the parallelogram were given differently, how would that affect the base?

Tip: Always ensure to factorize expressions properly to simplify solving equations.