QuestionA zoo has a newborn elephant weighing 200 pounds. Its weight increases by half the previous month's weight. Which equation helps find $t$ for 675 pounds, and what is $t$ ? Options: $200\left(\frac{3}{2}\right)^{t}=675$ ; Solutions: 2, 3, 4, or 6 months?

Studdy Solution

STEP 1

Assumptions1. The initial weight of the elephant is200 pounds. . The weight of the elephant increases exponentially each month by half the previous month's weight.
3. We are looking for the time it will take for the elephant to weigh675 pounds.

STEP 2

We need to set up an equation that models the weight of the elephant over time. The general form of an exponential equation is $y = ab^t$ where- $y$ is the final amount- $a$ is the initial amount- $b$ is the growth factor- $t$ is the time

STEP 3

In this case, the initial amount $a$ is the initial weight of the elephant, which is200 pounds. The final amount $y$ is the final weight of the elephant, which is675 pounds. The growth factor $b$ is1.5 (since the weight increases by half the previous month's weight, the growth factor is1 +0.5 =1.5). We are looking for the time $t$ .
So, the equation we need to solve is $675 =200 \times1.5^t$

STEP 4

Now we need to solve this equation for $t$ . First, divide both sides of the equation by200 to isolate the exponential term $\frac{675}{200} =1.^t$

STEP 5

Take the natural logarithm (ln) of both sides to remove the exponential term $\ln\left(\frac{675}{200}\right) = \ln\left(1.5^t\right)$

STEP 6

Use the property of logarithms that $\ln(a^b) = b \ln(a)$ to bring the $t$ down in front of the logarithm $\ln\left(\frac{675}{200}\right) = t \ln(1.5)$

STEP 7

Finally, divide both sides by $\ln(1.5)$ to solve for $t$ $t = \frac{\ln\left(\frac{675}{200}\right)}{\ln(1.5)}$

STEP 8

Calculate the value of $t$ $t = \frac{\ln\left(\frac{675}{200}\right)}{\ln(1.5)} \approx4.47$ Since the time must be an integer (as we can't have a fraction of a month), we round up to the nearest whole number, which gives us5 months.
So, the correct equation from the table is $200\left(\frac{3}{2}\right)^{t}=675$ and the correct solution from the table is5 months (which is not listed in the table).

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QuestionA zoo has a newborn elephant weighing 200 pounds. Its weight increases by half the previous month's weight. Which equation helps find $t$ for 675 pounds, and what is $t$ ? Options: $200\left(\frac{3}{2}\right)^{t}=675$ ; Solutions: 2, 3, 4, or 6 months?

Studdy Solution

STEP 1

Assumptions1. The initial weight of the elephant is200 pounds. . The weight of the elephant increases exponentially each month by half the previous month's weight.
3. We are looking for the time it will take for the elephant to weigh675 pounds.

STEP 2

We need to set up an equation that models the weight of the elephant over time. The general form of an exponential equation is $y = ab^t$ where- $y$ is the final amount- $a$ is the initial amount- $b$ is the growth factor- $t$ is the time

STEP 3

STEP 4

Now we need to solve this equation for $t$ . First, divide both sides of the equation by200 to isolate the exponential term $\frac{675}{200} =1.^t$

STEP 5

Take the natural logarithm (ln) of both sides to remove the exponential term $\ln\left(\frac{675}{200}\right) = \ln\left(1.5^t\right)$

STEP 6

Use the property of logarithms that $\ln(a^b) = b \ln(a)$ to bring the $t$ down in front of the logarithm $\ln\left(\frac{675}{200}\right) = t \ln(1.5)$

STEP 7

Finally, divide both sides by $\ln(1.5)$ to solve for $t$ $t = \frac{\ln\left(\frac{675}{200}\right)}{\ln(1.5)}$

STEP 8

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QuestionA zoo has a newborn elephant weighing 200 pounds. Its weight increases by half the previous month's weight. Which equation helps find ttt for 675 pounds, and what is ttt? Options: 200(32)t=675200\left(\frac{3}{2}\right)^{t}=675200(23​)t=675; Solutions: 2, 3, 4, or 6 months?

Studdy Solution

STEP 1

Assumptions1. The initial weight of the elephant is200 pounds. . The weight of the elephant increases exponentially each month by half the previous month's weight.3. We are looking for the time it will take for the elephant to weigh675 pounds.

STEP 2

We need to set up an equation that models the weight of the elephant over time. The general form of an exponential equation isy=abty = ab^ty=abtwhere- yyy is the final amount- aaa is the initial amount- bbb is the growth factor- ttt is the time

STEP 3

STEP 4

Now we need to solve this equation for ttt. First, divide both sides of the equation by200 to isolate the exponential term675200=1.t\frac{675}{200} =1.^t200675​=1.t

STEP 5

Take the natural logarithm (ln) of both sides to remove the exponential termln⁡(675200)=ln⁡(1.5t)\ln\left(\frac{675}{200}\right) = \ln\left(1.5^t\right)ln(200675​)=ln(1.5t)

STEP 6

Use the property of logarithms that ln⁡(ab)=bln⁡(a)\ln(a^b) = b \ln(a)ln(ab)=bln(a) to bring the ttt down in front of the logarithmln⁡(675200)=tln⁡(1.5)\ln\left(\frac{675}{200}\right) = t \ln(1.5)ln(200675​)=tln(1.5)

STEP 7

Finally, divide both sides by ln⁡(1.5)\ln(1.5)ln(1.5) to solve for tttt=ln⁡(675200)ln⁡(1.5)t = \frac{\ln\left(\frac{675}{200}\right)}{\ln(1.5)}t=ln(1.5)ln(200675​)​

STEP 8

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QuestionA zoo has a newborn elephant weighing 200 pounds. Its weight increases by half the previous month's weight. Which equation helps find $t$ for 675 pounds, and what is $t$ ? Options: $200\left(\frac{3}{2}\right)^{t}=675$ ; Solutions: 2, 3, 4, or 6 months?

Assumptions1. The initial weight of the elephant is200 pounds. . The weight of the elephant increases exponentially each month by half the previous month's weight.
3. We are looking for the time it will take for the elephant to weigh675 pounds.

We need to set up an equation that models the weight of the elephant over time. The general form of an exponential equation is $y = ab^t$ where- $y$ is the final amount- $a$ is the initial amount- $b$ is the growth factor- $t$ is the time

Now we need to solve this equation for $t$ . First, divide both sides of the equation by200 to isolate the exponential term $\frac{675}{200} =1.^t$

Take the natural logarithm (ln) of both sides to remove the exponential term $\ln\left(\frac{675}{200}\right) = \ln\left(1.5^t\right)$

Use the property of logarithms that $\ln(a^b) = b \ln(a)$ to bring the $t$ down in front of the logarithm $\ln\left(\frac{675}{200}\right) = t \ln(1.5)$

Finally, divide both sides by $\ln(1.5)$ to solve for $t$ $t = \frac{\ln\left(\frac{675}{200}\right)}{\ln(1.5)}$