Question

question
enter the missing value in the area model to find 5(2t + 4)
answer attempt 1 out of 2
according to the model above, 5(2t + 4) =
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Explanation:

Step1: Analyze the area model

The area model shows that one part is \( 3t \times \text{something} = 180 \)? Wait, no, looking at the model, we have a rectangle with one side \( 3t \) and the other part's area is 180? Wait, actually, the area model for \( 5(3t + 4) \) should be distributing 5 to both terms. Wait, maybe the model is a rectangle divided into two parts: one with length \( 3t \) and width 5, and the other with length 4 and width 5. Wait, no, the given model has \( 3t \) and 180? Wait, maybe first find the missing side. Wait, if one part is \( 3t \times 5 = 15t \)? No, wait, the area of the lower rectangle is 180, and the height is 5? Wait, no, the problem is to find \( 5(3t + 4) \) using the area model. Wait, maybe the area model has two rectangles: one with length \( 3t \) and width 5, and the other with length 4 and width 5. But the lower rectangle has area 180? Wait, maybe first find the width. Wait, if the lower rectangle has length 4 and area 180? No, that doesn't make sense. Wait, maybe the model is \( 5 \times (3t + 4) \), and the lower rectangle is \( 5 \times 4 = 20 \)? No, the given is 180. Wait, maybe I misread. Wait, the problem says "Enter the missing value in the area model to find \( 5(3t + 4) \)". Wait, maybe the area of the upper rectangle is \( 5 \times 3t = 15t \), and the lower rectangle is \( 5 \times 4 = 20 \)? But the given lower rectangle is 180? Wait, no, maybe the model is \( 3t \times \text{width} = 180 \), and the width is 5? Then \( 3t \times 5 = 15t \)? No, 3t*5=15t, not 180. Wait, maybe the width is 60? No, this is confusing. Wait, no, the correct way is to distribute 5 into \( 3t + 4 \). So \( 5(3t + 4) = 5 \times 3t + 5 \times 4 \).

Step2: Distribute the 5

Using the distributive property \( a(b + c) = ab + ac \), here \( a = 5 \), \( b = 3t \), \( c = 4 \). So \( 5 \times 3t = 15t \) and \( 5 \times 4 = 20 \). Wait, but the lower rectangle has area 180? Wait, maybe the model is different. Wait, maybe the height is 5, and the lower rectangle has length 4, so area is 54=20, but the given is 180. Wait, maybe there's a typo, but regardless, the algebraic expansion is \( 5(3t + 4) = 15t + 20 \). Wait, but maybe the area model has the lower rectangle as 54=20, but the given is 180. Wait, maybe I misread the model. Wait, the problem says "According to the model above, \( 5(3t + 4) = \)". So using the distributive property, \( 5(3t + 4) = 5 \times 3t + 5 \times 4 = 15t + 20 \). But maybe the model has the lower rectangle as 180, which would mean 5*4=20, but that's not 180. Wait, maybe the model is \( 3t \times 5 = 15t \) and \( 4 \times 5 = 20 \), but the given lower area is 180, which is wrong. Wait, maybe the model is \( 5 \times (3t + 4) \), and the missing value is the area of the upper rectangle, which is \( 5 \times 3t = 15t \), and the lower is \( 5 \times 4 = 20 \), but the given lower is 180. Wait, maybe the original problem has a different number. Wait, no, the correct expansion of \( 5(3t + 4) \) is \( 15t + 20 \). Maybe the 180 is a mistake, but following the distributive property, we distribute 5 to both terms.

Answer:

\( 15t + 20 \)