A biologist has found that the number of branches on a certain rare tree can be modeled by the polynomial b(y) = 4y² + y where y is the number of year

opatovaL

opatovaL

Answered question

2021-02-02

A biologist has found that the number of branches on a certain rare tree can be modeled by the polynomial b(y)=4y2+y where y is the number of years after the tree reaches a height of 6 feet. The number of leaves on each branch can be modeled by the polynomial l(y)=2y3+3y2+y. Write a polynomial describing the total number of leaves on the tree.

Answer & Explanation

avortarF

avortarF

Skilled2021-02-03Added 113 answers

The polynomial describing the total number of leaves on the tree is the product of the two models. This is like multiplying the number of branches by the number of trees. If t(y)t(y) represents the total number of leaves on the tree, we write:
t(y)=b(y)l(y)
t(y)=(4y2+y)(2y3+3y2+y)
Use distributive property:
t(y)=2y3(4y2+y)+3y2(4y2+y)+y(4y2+y)
t(y)=(8y5+2y4)+(12y4+3y3)+(4y3+y2)
t(y)=8y5+14y4+7y3+y2
RizerMix

RizerMix

Expert2023-06-18Added 656 answers

Step 1: Let's denote the total number of leaves on the tree as T(y). The expression for T(y) can be obtained by multiplying b(y) with l(y):
T(y)=b(y)·l(y)
Substituting the given polynomials, we have:
T(y)=(4y2+y)·(2y3+3y2+y)
Step 2: To simplify this expression, we need to distribute and combine like terms:
T(y)=8y5+12y4+4y3+2y4+3y3+y2
T(y)=8y5+14y4+7y3+y2
Therefore, the polynomial describing the total number of leaves on the tree is:
T(y)=8y5+14y4+7y3+y2
Don Sumner

Don Sumner

Skilled2023-06-18Added 184 answers

Result:
L(y)=8y5+14y4+7y3+y2
Solution:
Given that the number of branches on the tree is modeled by the polynomial b(y)=4y2+y, and the number of leaves on each branch is modeled by the polynomial l(y)=2y3+3y2+y, we can write the polynomial describing the total number of leaves as the product of these two polynomials.
Let's represent the total number of leaves on the tree as a polynomial L(y). We can express it using the polynomial multiplication operation:
L(y)=b(y)·l(y)
Substituting the given polynomial expressions for b(y) and l(y):
L(y)=(4y2+y)·(2y3+3y2+y)
To simplify the multiplication, we can use the distributive property and multiply each term in the first polynomial by each term in the second polynomial:
L(y)=4y2·2y3+4y2·3y2+4y2·y+y·2y3+y·3y2+y·y
Simplifying each term:
L(y)=8y5+12y4+4y3+2y4+3y3+y2
Combining like terms:
L(y)=8y5+14y4+7y3+y2
Therefore, the polynomial describing the total number of leaves on the tree is L(y)=8y5+14y4+7y3+y2.
Vasquez

Vasquez

Expert2023-06-18Added 669 answers

To find the polynomial describing the total number of leaves on the tree, we need to multiply the polynomial representing the number of branches, b(y)=4y2+y, by the polynomial representing the number of leaves on each branch, l(y)=2y3+3y2+y.
We can perform the multiplication by distributing each term of b(y) to every term of l(y) and then combining like terms.
Total number of leaves=b(y)·l(y)=(4y2+y)·(2y3+3y2+y)=8y5+12y4+4y3+2y4+3y3+y2=8y5+14y4+7y3+y2
Therefore, the polynomial describing the total number of leaves on the tree is 8y5+14y4+7y3+y2.

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