(PLS HURRY I'LL MARK BRAINLIEST)

Mathematics

1 answer:

ch4aika [34]2 years ago

8 0

Answer:

Answer: (0, -4) and (6, 1)

• x(t) = 6t --- equation (a)

• y(t) = 3t - 2 --- equation (b)

→ From equation (a), make t the subject:

$→ \: { \tt{x(t) = 6t }} \\ \\ → \: { \tt{t = \frac{x(t)}{6} }}$

• substitute for t in equation (b)

$→ \: { \tt{y(t) = 3(\frac{x(t)}{6}) - 2 }} \\ \\ → \: { \tt{y(t) = \frac{x(t)}{2} - 2 }}$

• Assume t is 1:

$→ \: { \tt{y = \frac{x}{2} - 2}} \\ \\ → \: { \boxed{ \tt{2y = x - 4}}}$

• when x is zero, y is -4

$→ \: { \tt{2y = 0 - 4 = - 4}} \\$

• when x is 6, y is 1

$→ \: { \tt{2y = 6 - 4 = 2 }} \\ { \tt{y = 1}}$

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A forestry company hopes to generate credits from the carbon sequestered in its plantations.The equation describing the forest w

Dmitry [639]

Answer:

Explanation:

Given:

The equation describing the forest wood biomass per hectare as a function of plantation age t is:

y(t) = 5 + 0.005t^2 + 0.024t^3 − 0.0045t^4

The equation that describes the annual growth in wood biomass is:

y ′ (t) = 0.01t + 0.072t^2 - 0.018t^3

To find:

a) The year the annual growth achieved its highest possible value

b) when does y ′ (t) achieve its highest value?

To determine the year the highest possible value was achieved, we will set the derivative y'(t) to zero. The values of t will be substituted into the second derivative to get the highest value

$\begin{gathered} 0\text{ = 0.01t + 0.072t}^2\text{ - 0.018t}^3 \\ using\text{ a graphing tool,} \\ t\text{ = -0.1343} \\ \text{t = 0} \\ t\text{ = 4.1343} \\ \\ Since\text{ we can't have time as a negative value, t = 0 or t = 4.1343} \end{gathered}$ $\begin{gathered} To\text{ ascertain which is the maximum, we take a second derivative} \\ y^{\prime}\text{'\lparen t\rparen = 0.01 + 0.144t - 0.054t}^2 \\ \\ substitute\text{ t = 0 and t = 4.13 respectively} \\ when\text{ t = 0 } \\ y^{\prime}\text{'\lparen t\rparen= 0.01 + 0.144\lparen0\rparen - 0.054\lparen0\rparen}^2 \\ y^{\prime}\text{'\lparen t\rparen= 0.01 + 0 - 0} \\ y^{\prime}\text{'\lparen t\rparen= 0.01 > 0} \\ \\ y^{\prime}\text{'\lparen t\rparen= 0.01 + 0.144\lparen4.13\rparen - 0.054\lparen4.13\rparen}^2 \\ y^{\prime}\text{'\lparen t\rparen= -0.316 < 0} \\ \\ if\text{ }y^{\prime}\text{'\lparen t\rparen > 0 , then it is minimum, } \\ if\text{ }y^{\prime}\text{'\lparen t\rparen < 0, then it is maximum} \end{gathered}$

SInce t = 4.13, gives y ′' (t) = -0.316 (< 0). This makes it the maximum value of t

The year the annual growth achieved its highest possible value to the nearest whole number will be

year 4

b) y ′ (t) will achieve its highest value, when we substitute the value of t that gives into the initial function.

Initial function: y(t) = 5 + 0.005t^2 + 0.024t^3 − 0.0045t^4