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Math  /  Geometry

QuestionIllustrative Mathernat
8. To grow properly, each tomato plant needs 1.5 square feet of soil and each broccoli plant needs 2.25 square feet of soil. The graph shows the different combinations of broccoli and tomato plants in an 18 square foot plot of soil.
Match each point to the statement that describes it. A. Point A
1. The soil is fully used when 6 tomato B. Point B plants and 4 broccoli plants are planted. C. Point C D. Point D
2. Only broccoli was planted, but the plot is fully used and all plants can grow properly.
3. After 3 tomato plants and 2 broccoli plants were planted, there is still extra space in the plot.
4. With 4 tomato plants and 6 broccoli plants planted, the plot is overcrowded.

Studdy Solution

STEP 1

What is this asking? We need to figure out which combinations of tomato and broccoli plants will perfectly fill, under-fill, or over-fill an 18 square foot garden! Watch out! Don't forget each plant needs a specific amount of space, and mixing them up will lead to wrong answers.

STEP 2

1. Calculate space needed for each point.
2. Match points to statements.

STEP 3

Alright, for Point A, we have **zero** tomato plants, so they take up 0 tomatoes⋅1.5sq fttomato=0 sq ft\text{0 tomatoes} \cdot 1.5 \frac{\text{sq ft}}{\text{tomato}} = 0 \text{ sq ft}.
We've got **8** broccoli plants, needing 2.252.25 square feet each.
So, they'll use 8 broccoli⋅2.25sq ftbroccoli=18 sq ft8 \text{ broccoli} \cdot 2.25 \frac{\text{sq ft}}{\text{broccoli}} = 18 \text{ sq ft}.
That's **18 sq ft** total!

STEP 4

For Point B, we've got **6** tomato plants taking up 6 tomatoes⋅1.5sq fttomato=9 sq ft6 \text{ tomatoes} \cdot 1.5 \frac{\text{sq ft}}{\text{tomato}} = 9 \text{ sq ft}.
Plus **4** broccoli plants needing 4 broccoli⋅2.25sq ftbroccoli=9 sq ft4 \text{ broccoli} \cdot 2.25 \frac{\text{sq ft}}{\text{broccoli}} = 9 \text{ sq ft}.
Adding those together, 9 sq ft+9 sq ft=18 sq ft9 \text{ sq ft} + 9 \text{ sq ft} = 18 \text{ sq ft} total!

STEP 5

Point C has **8** tomato plants, gobbling up 8 tomatoes⋅1.5sq fttomato=12 sq ft8 \text{ tomatoes} \cdot 1.5 \frac{\text{sq ft}}{\text{tomato}} = 12 \text{ sq ft}.
And **2** broccoli plants using 2 broccoli⋅2.25sq ftbroccoli=4.5 sq ft2 \text{ broccoli} \cdot 2.25 \frac{\text{sq ft}}{\text{broccoli}} = 4.5 \text{ sq ft}.
Together, that's 12 sq ft+4.5 sq ft=16.5 sq ft12 \text{ sq ft} + 4.5 \text{ sq ft} = 16.5 \text{ sq ft}.

STEP 6

Lastly, Point D has **4** tomato plants needing 4 tomatoes⋅1.5sq fttomato=6 sq ft4 \text{ tomatoes} \cdot 1.5 \frac{\text{sq ft}}{\text{tomato}} = 6 \text{ sq ft}.
And **6** broccoli plants needing 6 broccoli⋅2.25sq ftbroccoli=13.5 sq ft6 \text{ broccoli} \cdot 2.25 \frac{\text{sq ft}}{\text{broccoli}} = 13.5 \text{ sq ft}.
In total, 6 sq ft+13.5 sq ft=19.5 sq ft6 \text{ sq ft} + 13.5 \text{ sq ft} = 19.5 \text{ sq ft}.

STEP 7

This matches Point B exactly, which uses **18 sq ft**, perfect!

STEP 8

This is Point A, with **8 broccoli** plants using all **18 sq ft** and no tomatoes!

STEP 9

We didn't calculate this exactly, but Point C, with 8 tomatoes and 2 broccoli, *almost* fills the plot at 16.5 sq ft.
Having *fewer* plants will definitely leave extra space.
Let's check: 3 tomatoes take up 3⋅1.5=4.53 \cdot 1.5 = 4.5 sq ft, and 2 broccoli take up 2⋅2.25=4.52 \cdot 2.25 = 4.5 sq ft, for a total of 4.5+4.5=94.5 + 4.5 = 9 sq ft.
Plenty of space left over!

STEP 10

This matches Point D, which needs **19.5 sq ft**, more than the available **18 sq ft**!

STEP 11

A - 2 B - 1 C - 3 D - 4

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