Respuesta :
Answer:
a) [tex] P_5 = \frac{5^3 +5(5) +6}{6}=\frac{156}{6}=26[/tex]
So the would be the maximum number of pieces for this case.
b) [tex] \frac{n^3 +5n +6}{6} \geq 60[/tex]
And we can solve for n like this:
[tex] n^3 +5n +6 \geq 360[/tex]
[tex] n(n^2+5) \geq 354[/tex]
And we can see if for some values of n the condition is satisfied:
n =6 [tex] 6(6^2+5)=246 <354[/tex] Not satisfied
n =7 [tex] 7(7^2+5)=378 \geq 354[/tex] Satisfied
n =8 [tex] 8(8^2+5)=552 \geq 354[/tex] Satisfied
So then our answer should be 7 or 8 but as we can see:
[tex] P_7 = \frac{7^3 +5(7) +6}{6}=\frac{384}{6}=64[/tex]
[tex] P_8 = \frac{8^3 +5(8) +6}{6}=\frac{558}{6}=93[/tex]
So the maximum value that satisfy the requirement is n=7
Step-by-step explanation:
Assuming this complete problem: "One straight cut through a thick piece of cheese produces two pieces. Two straight cuts can produce a maximum of 4 pieces. Three straight cuts can produce a maximum of 8 pieces. You might be inclined to think that every additional cut doubles the previous number of pieces. However, for four straight cuts, you will find that you get a maximum of 15 pieces. An nth-term formula for the maximum number of pieces,
[tex]P_n[/tex], that can be produced by n straight cuts is":
[tex] P_n = \frac{n^3 +5n +6}{6}[/tex]
. a. Use the nth-term formula to determine the maximum number of pieces that can be produced by five straight cuts.
b. What is the smallest number of straight cuts that you can use if you wish to produce at least 60 pieces? Hint: Use the nth-term formula and experiment with larger and larger values of n."
Part a
For this case we jusy need to replace n=5 in the nth term formula and we got:
[tex] P_5 = \frac{5^3 +5(5) +6}{6}=\frac{156}{6}=26[/tex]
So the would be the maximum number of pieces for this case.
Part b
So for this case we need to look for a value of n such that we produce at least 60 pieces, so we want to solve this:
[tex] \frac{n^3 +5n +6}{6} \geq 60[/tex]
And we can solve for n like this:
[tex] n^3 +5n +6 \geq 360[/tex]
[tex] n(n^2+5) \geq 354[/tex]
And we can see if for some values of n the condition is satisfied:
n =6 [tex] 6(6^2+5)=246 <354[/tex] Not satisfied
n =7 [tex] 7(7^2+5)=378 \geq 354[/tex] Satisfied
n =8 [tex] 8(8^2+5)=552 \geq 354[/tex] Satisfied
So then our answer should be 7 or 8 but as we can see:
[tex] P_7 = \frac{7^3 +5(7) +6}{6}=\frac{384}{6}=64[/tex]
[tex] P_8 = \frac{8^3 +5(8) +6}{6}=\frac{558}{6}=93[/tex]
So the maximum value that satisfy the requirement is n=7
The maximum number of pieces for 5 cuts is 26
How to determine the maximum number of pieces for 5 cuts?
The number of pieces for n cuts is given as:
[tex]P(n) = \frac{n^3 + 5n + 6}6[/tex]
Substitute 5 for n
[tex]P(5) = \frac{5^3 + 5*5 + 6}6[/tex]
Evaluate the sum
[tex]P(5) = \frac{156}6[/tex]
Evaluate the quotient
[tex]P(5) = 26[/tex]
Hence, the maximum number of pieces for 5 cuts is 26
Read more about sequence at:
https://brainly.com/question/6561461
