Question Find the charity's initial budget and a function for the trees planted weekly, given they planted 71 trees in week 2 and 122 in week 5.

Studdy Solution

STEP 1

Assumptions
1. The number of trees planted is proportional to the amount of money raised.
2. The charity plants one tree for every dollar they raise.
3. The charity has been receiving donations at a steady pace.
4. The charity planted 71 trees on week 2 and 122 trees on week 5.

STEP 2

Let's denote the number of trees planted as $T$ , the week number as $w$ , and the initial budget as $B$ . We are looking for a linear function of the form $T = m \cdot w + B$ , where $m$ is the slope of the line.

STEP 3

We can find the slope $m$ by using the formula for the slope of a line, which is the change in $T$ divided by the change in $w$ .
$m = \frac{T_2 - T_1}{w_2 - w_1}$

STEP 4

Now, plug in the given values for $T_2$ , $T_1$ , $w_2$ , and $w_1$ to calculate the slope.
$m = \frac{122 - 71}{5 - 2}$

STEP 5

Calculate the slope.
$m = \frac{51}{3} = 17$

STEP 6

Now that we have the slope, we can find the initial budget $B$ by rearranging the equation $T = m \cdot w + B$ to $B = T - m \cdot w$ .

STEP 7

Plug in the values for $T$ , $m$ , and $w$ from week 2 to calculate the initial budget.
$B = 71 - 17 \cdot 2$

STEP 8

Calculate the initial budget.
$B = 71 - 34 = 37$

STEP 9

Now that we have the initial budget and the slope, we can construct the function $A(w)$ for the number of trees they planted every week.
$A(w) = 17w + 37$
The initial budget of the charity was $37 and the function for the number of trees they planted every week is$ A(w) = 17w + 37$.

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Question Find the charity's initial budget and a function for the trees planted weekly, given they planted 71 trees in week 2 and 122 in week 5.

Studdy Solution

STEP 1

Assumptions
1. The number of trees planted is proportional to the amount of money raised.
2. The charity plants one tree for every dollar they raise.
3. The charity has been receiving donations at a steady pace.
4. The charity planted 71 trees on week 2 and 122 trees on week 5.

STEP 2

Let's denote the number of trees planted as $T$ , the week number as $w$ , and the initial budget as $B$ . We are looking for a linear function of the form $T = m \cdot w + B$ , where $m$ is the slope of the line.

STEP 3

We can find the slope $m$ by using the formula for the slope of a line, which is the change in $T$ divided by the change in $w$ .
$m = \frac{T_2 - T_1}{w_2 - w_1}$

STEP 4

Now, plug in the given values for $T_2$ , $T_1$ , $w_2$ , and $w_1$ to calculate the slope.
$m = \frac{122 - 71}{5 - 2}$

STEP 5

Calculate the slope.
$m = \frac{51}{3} = 17$

STEP 6

Now that we have the slope, we can find the initial budget $B$ by rearranging the equation $T = m \cdot w + B$ to $B = T - m \cdot w$ .

STEP 7

Plug in the values for $T$ , $m$ , and $w$ from week 2 to calculate the initial budget.
$B = 71 - 17 \cdot 2$

STEP 8

Calculate the initial budget.
$B = 71 - 34 = 37$

STEP 9

Now that we have the initial budget and the slope, we can construct the function $A(w)$ for the number of trees they planted every week.
$A(w) = 17w + 37$
The initial budget of the charity was $37 and the function for the number of trees they planted every week is$ A(w) = 17w + 37$.

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Question Find the charity's initial budget and a function for the trees planted weekly, given they planted 71 trees in week 2 and 122 in week 5.

Studdy Solution

STEP 1

Assumptions1. The number of trees planted is proportional to the amount of money raised.2. The charity plants one tree for every dollar they raise.3. The charity has been receiving donations at a steady pace.4. The charity planted 71 trees on week 2 and 122 trees on week 5.

STEP 2

Let's denote the number of trees planted as TTT, the week number as www, and the initial budget as BBB. We are looking for a linear function of the form T=m⋅w+BT = m \cdot w + BT=m⋅w+B, where mmm is the slope of the line.

STEP 3

We can find the slope mmm by using the formula for the slope of a line, which is the change in TTT divided by the change in www.m=T2−T1w2−w1m = \frac{T_2 - T_1}{w_2 - w_1}m=w2​−w1​T2​−T1​​

STEP 4

Now, plug in the given values for T2T_2T2​, T1T_1T1​, w2w_2w2​, and w1w_1w1​ to calculate the slope.m=122−715−2m = \frac{122 - 71}{5 - 2}m=5−2122−71​

STEP 5

Calculate the slope.m=513=17m = \frac{51}{3} = 17m=351​=17

STEP 6

Now that we have the slope, we can find the initial budget BBB by rearranging the equation T=m⋅w+BT = m \cdot w + BT=m⋅w+B to B=T−m⋅wB = T - m \cdot wB=T−m⋅w.

STEP 7

Plug in the values for TTT, mmm, and www from week 2 to calculate the initial budget.B=71−17⋅2B = 71 - 17 \cdot 2B=71−17⋅2

STEP 8

Calculate the initial budget.B=71−34=37B = 71 - 34 = 37B=71−34=37

STEP 9

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Assumptions
1. The number of trees planted is proportional to the amount of money raised.
2. The charity plants one tree for every dollar they raise.
3. The charity has been receiving donations at a steady pace.
4. The charity planted 71 trees on week 2 and 122 trees on week 5.

Let's denote the number of trees planted as $T$ , the week number as $w$ , and the initial budget as $B$ . We are looking for a linear function of the form $T = m \cdot w + B$ , where $m$ is the slope of the line.

We can find the slope $m$ by using the formula for the slope of a line, which is the change in $T$ divided by the change in $w$ .
$m = \frac{T_2 - T_1}{w_2 - w_1}$

Now, plug in the given values for $T_2$ , $T_1$ , $w_2$ , and $w_1$ to calculate the slope.
$m = \frac{122 - 71}{5 - 2}$

Calculate the slope.
$m = \frac{51}{3} = 17$

Now that we have the slope, we can find the initial budget $B$ by rearranging the equation $T = m \cdot w + B$ to $B = T - m \cdot w$ .

Plug in the values for $T$ , $m$ , and $w$ from week 2 to calculate the initial budget.
$B = 71 - 17 \cdot 2$

Calculate the initial budget.
$B = 71 - 34 = 37$