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Given the notation and potential context, the expression seems to involve a function [tex]\( r(t) \)[/tex]. Assuming the problem intends to define a function [tex]\( r(t) \)[/tex] or solve for [tex]\( t \)[/tex] in terms of some variable, let's reformat it to make more sense:

Define the function [tex]\( r(t) \)[/tex]:

[tex]\[ r(t) = 4t^4 + 3r^3 - t^2 + 6at \][/tex]

where [tex]\( t \in \mathbb{R} \)[/tex].

If the task was intended to solve for [tex]\( t \)[/tex] or to explore properties of the function, further instructions would be necessary. For now, the statement above clearly defines the function [tex]\( r(t) \)[/tex].

Sagot :

GinnyAnswer
Certainly! Let’s work through the problem step-by-step, making sure to break it down in an easy-to-follow manner.

The given equation is:

[tex]\[ r(t) = 4t^4 + 3r^3 - t^2 + 6at \][/tex]
where [tex]\( t = p \)[/tex].

This means that we need to evaluate the function [tex]\( r(t) \)[/tex] at [tex]\( t = p \)[/tex]. Let's rewrite the function by substituting [tex]\( t \)[/tex] with [tex]\( p \)[/tex]:

[tex]\[ r(p) = 4p^4 + 3r^3 - p^2 + 6ap \][/tex]

Let's analyze this equation in detail:

1. Substitute [tex]\( t \)[/tex] with [tex]\( p \)[/tex] in the function [tex]\( r(t) \)[/tex] to get [tex]\( r(p) \)[/tex]:

[tex]\[ r(p) = 4(p)^4 + 3r^3 - (p)^2 + 6a(p) \][/tex]

2. Simplify each term:
- For [tex]\( 4(p)^4 \)[/tex], it remains [tex]\( 4p^4 \)[/tex]
- The second term is [tex]\( 3r^3 \)[/tex], which doesn't change
- The third term, [tex]\((p)^2\)[/tex], simplifies to [tex]\( p^2 \)[/tex]
- The fourth term, [tex]\( 6a(p) \)[/tex], simplifies to [tex]\( 6ap \)[/tex]

3. Combine all these simplified terms together:

[tex]\[ r(p) = 4p^4 + 3r^3 - p^2 + 6ap \][/tex]

Hence, the evaluated function at [tex]\( t = p \)[/tex] is:

[tex]\[ r(p) = 4p^4 + 3r^3 - p^2 + 6ap \][/tex]

This completes the evaluation of the function [tex]\( r(t) \)[/tex] at [tex]\( t = p \)[/tex].
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