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elena-14-01-66 [18.8K]
3 years ago
10

Evaluate the integral. 3 2 t3i t t − 2 j t sin(πt)k dt

Mathematics
1 answer:
sveta [45]3 years ago
3 0

∫(t = 2 to 3) t^3 dt

= (1/4)t^4 {for t = 2 to 3}

= 65/4.

----

∫(t = 2 to 3) t √(t - 2) dt

= ∫(u = 0 to 1) (u + 2) √u du, letting u = t - 2

= ∫(u = 0 to 1) (u^(3/2) + 2u^(1/2)) du

= [(2/5) u^(5/2) + (4/3) u^(3/2)] {for u = 0 to 1}

= 26/15.

----

For the k-entry, use integration by parts with

u = t, dv = sin(πt) dt

du = 1 dt, v = (-1/π) cos(πt).


So, ∫(t = 2 to 3) t sin(πt) dt

= (-1/π) t cos(πt) {for t = 2 to 3} - ∫(t = 2 to 3) (-1/π) cos(πt) dt

= (-1/π) (3 * -1 - 2 * 1) + [(1/π^2) sin(πt) {for t = 2 to 3}]

= 5/π + 0

= 5/π.

Therefore,

∫(t = 2 to 3) <t^3, t√(t - 2), t sin(πt)> dt = <65/4, 26/15, 5/π>.

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John runs a computer software store. Yesterday he counted 123 people who walked by the store, 56 of whom came into the store. Of
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Answer:

a) There is a 45.53% probability that a person who walks by the store will enter the store.

b) There is a 41.07% probability that a person who walks into the store will buy something.

c) There is a 18.70% probability that a person who walks by the store will come in and buy something.

d) There is a 58.93% probability that a person who comes into the store will buy nothing.

Step-by-step explanation:

This a probability problem.

The probability formula is given by:

P = \frac{D}{T}

In which P is the probability, D is the number of desired outcomes and T is the number of total outcomes.

The problem states that:

123 people walked by the store.

56 people came into the store.

23 bought something in the store.

(a) Estimate the probability that a person who walks by the store will enter the store.

123 people walked by the store and 56 entered the store, so T = 123, D = 56.

So

P = \frac{D}{T} = \frac{56}{123} = 0.4553

There is a 45.53% probability that a person who walks by the store will enter the store.

(b) Estimate the probability that a person who walks into the store will buy something.

56 people came into the store and 23 bought something, so T = 56, D = 23.

So

P = \frac{D}{T} = \frac{23}{56} = 0.4107

There is a 41.07% probability that a person who walks into the store will buy something.

(c) Estimate the probability that a person who walks by the store will come in and buy something.

123 people walked by the store and 23 came in and bought something, so T = 123, D = 23.

So

P = \frac{D}{T} = \frac{23}{123} = 0.1870

There is a 18.70% probability that a person who walks by the store will come in and buy something.

(d) Estimate the probability that a person who comes into the store will buy nothing.

Of the 56 people whom came into the store, 23 bought something. This means that 56-23 = 33 of them did not buy anything. So:

D = 33, T = 56

P = \frac{D}{T} = \frac{33}{56} = 0.5893

There is a 58.93% probability that a person who comes into the store will buy nothing.

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