Question

Please help me with the 2 math questions and please include an explanation as well. Thank you!

I will delete answers that incomplete or has no explanation.

Answer 1

13) 4.9 m

14) 0.9 m

Explanation:

Question 13

The given diagram shows the height of the same cactus plant a year apart:

• Year 1 height = 1.6 m
• Year 2 height = 2 m

We are told that the cactus continues to grow at the same percentage rate. To calculate the growth rate per year (percentage increase), use the percentage increase formula:

`\begin{aligned}\sf Percentage \; increase &= (\sf Final\; value - Initial \;value)/(\sf Initial \;value)\\\\&=( 2-1.6)/(1.6)\\\\&=(0.4)/(1.6)\\\\&=0.25\end{aligned}`

Therefore, the growth rate of the height of the cactus is 25% per year.

As the cactus grows at a constant rate, we can use the exponential growth formula to calculate its height in Year 6.

`\boxed{\begin{minipage}{7.5 cm}\underline{Exponential Growth Formula}\\\\$y=a(1+r)^t$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value. \\ \phantom{ww}$\bullet$ $r$ is the growth factor (in decimal form).\\ \phantom{ww}$\bullet$ $t$ is the number of time periods.\\\end{minipage}}`

The initial value is the height in Year 1, so a = 1.6.

The growth factor is 25%, so r = 0.25.

As we wish to calculate its height in Year 6, the value of t is t = 5 (since there are 5 years between year 1 and year 6).

Substitute these values into the formula and solve for y (the height of the cactus):

`\begin{aligned}y&=a(1+r)^t\\&=1.6(1+0.25)^5\\&=1.6(1.25)^5\\&=1.6(3.0517578125)\\&=4.8828125\\&=4.9\; \sf m\;(nearest\;tenth)\end{aligned}`

Therefore, if the cactus continues to grow at the same rate, its height in Year 6 will be 4.9 meters (to the nearest tenth).

Check by multiplying the height each year by 1.25:

• Year 1 = 1.6 m
• Year 2 = 1.6 × 1.25 = 2 m
• Year 3 = 2 × 1.25 = 2.5 m
• Year 4 = 2.5 × 1.25 = 3.125 m
• Year 5 = 3.125 × 1.25 = 3.09625 m
• Year 6 = 3.09625 × 1.25 = 4.8828125 m

`\hrulefill`

Question 14

The given diagram shows the height of the same snowman an hour apart:

• Initial height = 1.8 m
• Height after an hour = 1.53 m

We are told that the snowman continues to melt at the same percentage rate. To calculate the decay rate per hour (percentage decrease), use the percentage decrease formula:

`\begin{aligned}\sf Percentage \; decrease&= (\sf Initial\; value - Final\;value)/(\sf Initial \;value)\\\\&=(1.8-1.53)/(1.8)\\\\&=(0.27)/(1.8)\\\\&=0.15\end{aligned}`

Therefore, the decay rate of the snowman's height is 15% per hour.

As the snowman melts at a constant rate, we can use the exponential decay formula to calculate its height after another 3 hours.

`\boxed{\begin{minipage}{7.5 cm}\underline{Exponential Decay Formula}\\\\$y=a(1-r)^t$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the initial value. \\ \phantom{ww}$\bullet$ $r$ is the decay factor (in decimal form).\\ \phantom{ww}$\bullet$ $t$ is the number of time periods.\\\end{minipage}}`

The initial value is the snowman's initial height, so a = 1.8.

The decay factor is 15%, so r = 0.15.

As we wish to calculate the snowman's height after another 3 hours, the value of t is t = 4 (i.e. the first hour plus a further 3 hours).

Substitute these values into the formula and solve for y (the height of the snowman):

`\begin{aligned}y&=a(1-r)^t\\&=1.8(1-0.15)^4\\&=1.8(0.85)^4\\&=1.8(0.5220065)\\&=0.93961125\\&=0.9\; \sf m\;(nearest\;tenth)\end{aligned}`

Therefore, if the snowman continues to melt at the same rate, its height after another 3 hours will be 0.9 meters (to the nearest tenth).

Check by multiplying the height each hour by 0.85:

• Initial height = 1.8 m
• Height after 1 hour = 1.8 × 0.85 = 1.53
• Height after 2 hours = 1.53 × 0.85 = 1.3005
• Height after 3 hours = 1.3005 × 0.85 = 1.105425
• Height after 4 hours = 1.105425 × 0.85 = 0.93961125