**1. The floor of a rectangular cage has a length 4 feet greater than its width, \( w \). James will increase both dimensions of the floor by 2 feet. Which equation represents the new area, \( N \), of the floor of the cage?**
A. \( N = w^2 + 4w \)
B. \( N = w^2 + 6w \)
C. \( N = w^2 + 6w + 8 \)
D. \( N = w^2 + 8w + 12 \)
**2. Two boys, Shawn and Curtis, went for a walk. Shawn began walking 20 seconds earlier than Curtis. Shawn walked at a speed of 5 feet per second. Curtis walked at a speed of 6 feet per second. For how many seconds had Shawn been walking at the moment when the two boys had walked exactly the same distance?**
Answer (2)
The **equation **that represents the new area , N, of the floor of the cage is D) N = w² + 8w + 12.
Shawn had been **walking **for 120 seconds.
What is algebraic Expression?
Any **mathematical statement **that includes numbers, variables, and an arithmetic operation between them is known as an **expression **or algebraic expression. In the phrase 4m + 5, for instance, the terms 4m and 5 are separated from the variable m by the arithmetic sign +.
The **original area **of the rectangular cage is:
A = lw, where l is the length and w is the width.
We are given that the length of the floor is 4 feet greater than its width, so we can write l = w + 4.
The **new dimensions **of the floor are w + 2 and (w + 4) + 2 = w + 6. Therefore, the new area of the floor is:
N = (w + 2)(w + 6) = w² + 6w + 2w + 12 = w² + 8w + 12
Let t be the time in seconds that Curtis has been walking. Then Shawn has been walking for t + 20 seconds.
The **distance **that Curtis has walked is d = 6t, and the **distance **that Shawn has walked is d = 5(t + 20), since Shawn started 20 seconds earlier and walks at a slower pace.
We want to find the time when they have walked the same distance, so we set the two expressions for d **equal **to each other:
6t = 5(t + 20)
**Simplifying **and solving for t, we get:
6t = 5t + 100
t = 100
Therefore, Curtis had been walking for 100 seconds when the two boys had walked exactly the same distance . At that time, Shawn had been walking for t + 20 = 120 seconds.
Learn more about algebraic **Expression **here:
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The equation that represents the new area of the floor of the cage is N = w 2 + 8 w + 12 . Shawn was walking for 120 seconds when he and Curtis had walked the same distance. Thus, the answers are A for the first question and 120 seconds for the second question.
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