a bee flies at $20$ feet per second directly to a flowerbed from its hive. The bee stays at the flowerbed for $12$ minutes, and then flies directly back to the hive at $12$ feet per second. It is away from the hive for a total of $17$ minutes. What equation can you use to find the distance of the flowerbed from the hive?

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Solve equations with variables on both sides: word problems

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Q. a bee flies at

20

feet per second directly to a flowerbed from its hive. The bee stays at the flowerbed for

12

minutes, and then flies directly back to the hive at

12

feet per second. It is away from the hive for a total of

17

minutes. What equation can you use to find the distance of the flowerbed from the hive?

Convert time to seconds: Convert the total time the bee is away from the hive into seconds. $\newline$ The bee is away for a total of $17$ minutes. Since there are $60$ seconds in a minute, we multiply $17$ minutes by $60$ seconds per minute to get the total time in seconds. $\newline$ $17$ minutes $\times$ $60$ seconds/minute $=$ $1020$ seconds.
Calculate time at flowerbed: Convert the time the bee stays at the flowerbed into seconds. $\newline$ The bee stays at the flowerbed for $12$ minutes. Again, we multiply by $60$ seconds per minute to convert this time into seconds. $\newline$ $12$ minutes $\times 60$ seconds/minute $= 720$ seconds.
Calculate time spent flying: Calculate the time spent flying. $\newline$ The total time the bee is away includes the time spent at the flowerbed and the time spent flying to and from the flowerbed. To find the time spent flying, we subtract the time spent at the flowerbed from the total time away. $\newline$ Total time flying $= \text{Total time away} - \text{Time at flowerbed}$ $\newline$ Total time flying $= 1020 \text{ seconds} - 720 \text{ seconds} = 300 \text{ seconds}.$
Find distance to flowerbed: Let the distance to the flowerbed be represented by the variable $d$ (in feet). We need to find this distance, $d$ , which the bee travels at two different speeds: $20$ feet per second on the way to the flowerbed and $12$ feet per second on the way back.
Equation for distance to flowerbed: Write the equation for the distance traveled to the flowerbed. $\newline$ The bee flies to the flowerbed at $20$ feet per second. Let $t_1$ be the time in seconds it takes to get to the flowerbed. The distance $d$ can be expressed as: $\newline$ $d = 20 \times t_1$
Equation for distance from flowerbed: Write the equation for the distance traveled from the flowerbed back to the hive. $\newline$ The bee flies back to the hive at $12$ feet per second. Let $t_2$ be the time in seconds it takes to get back to the hive. The distance $d$ is the same, so it can also be expressed as: $\newline$ $d = 12 \times t_2$
Combine time equations: Since the bee flies directly to the flowerbed and then directly back, the times $t_1$ and $t_2$ must add up to the total flying time. $\newline$ Therefore, we have the equation: $\newline$ $t_1 + t_2 = 300$ seconds
Express time in terms of distance: Express $t_1$ and $t_2$ in terms of $d$ using the equations from Steps $5$ and $6$ . $\newline$ From Step $5$ : $t_1 = \frac{d}{20}$ $\newline$ From Step $6$ : $t_2 = \frac{d}{12}$
Substitute time expressions: Substitute the expressions for $t_1$ and $t_2$ into the equation from Step $7$ . $\newline$ $\frac{d}{20} + \frac{d}{12} = 300$
Find common denominator: Find a common denominator to combine the terms on the left side of the equation. $\newline$ The common denominator for $20$ and $12$ is $60$ . We rewrite the equation as: $\newline$ $(3d / 60) + (5d / 60) = 300$
Combine terms: Combine the terms on the left side of the equation. $\newline$ $(3d + 5d) / 60 = 300$ $\newline$ $8d / 60 = 300$
Solve for distance: Multiply both sides of the equation by $60$ to solve for $d$ . $\newline$ $8d = 300 \times 60$ $\newline$ $8d = 18000$
Solve for distance: Multiply both sides of the equation by $60$ to solve for $d$ . $\newline$ $8d = 300 \times 60$ $\newline$ $8d = 18000$ Divide both sides of the equation by $8$ to find the value of $d$ . $\newline$ $d = 18000 / 8$ $\newline$ $d = 2250$