Just to ensure I'm doing this method right, (for clarification)

If a = pi +3j - 7k, b = pi - pj +4k and the angle between a and is acute then the possible values for p are given by

PIT_PIT [208]

Answer:

The family of possible values for $p$ are:

$(-\infty, -4) \,\cup \,(7, +\infty)$

Step-by-step explanation:

By Linear Algebra, we can calculate the angle by definition of dot product:

$\cos \theta = \frac{\vec a\,\bullet\,\vec b}{\|\vec a\|\cdot \|\vec b\|}$ (1)

Where:

$\theta$ - Angle between vectors, in sexagesimal degrees.

$\|\vec a\|, \|\vec b \|$ - Norms of vectors $\vec {a}$ and $\vec{b}$

If $\theta$ is acute, then the cosine function is bounded between 0 a 1 and if we know that $\vec {a} = (p, 3, -7)$ and $\vec {b} = (p, -p, 4)$ , then the possible values for $p$ are:

Minimum ( $\cos \theta = 0$ )

$\frac{p^{2}-3\cdot p -28}{\sqrt{p^{2}+58}\cdot \sqrt{2\cdot p^{2}+16}} > 0$

Maximum ( $\cos \theta = 1$ )

$\frac{p^{2}-3\cdot p -28}{\sqrt{p^{2}+58}\cdot \sqrt{2\cdot p^{2}+16}} < 1$

With the help of a graphing tool we get the family of possible values for $p$ are:

$(-\infty, -4) \,\cup \,(7, +\infty)$

7 0

2 years ago

A cylinder has radius r and height h. A. How many times greater is the surface area of a cylinder when both dimensions are multi

Ierofanga [76]

Answer: A. Factor 2 => 4x greater

Factor 3 => 9x greater

Factor 5 => 25x greater

Step-by-step explanation: A. A cylinder is formed by 2 circles and a rectangle in the middle. That's why surface area is given by circumference of a circle, which is the length of the rectangle times height of the rectangle, i.e.:

A = 2.π.r.h

A cylinder of radius r and height h has area:

$A_{1}$ = 2πrh

If multiply both dimensions by a factor of 2:

$A_{2}$ = 2.π.2r.2h

$A_{2}$ = 8πrh

Comparing $A_{1}$ to $A_{2}$ :

$\frac{A_{2}}{A_{1}}$ = $\frac{8.\pi.rh}{2.\pi.rh}$ = 4

Doubling radius and height creates a surface area of a cylinder 4 times greater.

By factor 3:

$A_{3} = 2.\pi.3r.3h$

$A_{3} = 18.\pi.r.h$

Comparing areas:

$\frac{A_{3}}{A_{1}}$ = $\frac{18.\pi.r.h}{2.\pi.r.h}$ = 9

Multiplying by 3, gives an area 9 times bigger.

By factor 5:

$A_{5} = 2.\pi.5r.5h$

$A_{5} = 50.\pi.r.h$

Comparing:

$\frac{A_{5}}{A_{1}}$ = $\frac{50.\pi.r.h}{2.\pi.r.h}$ = 25

The new area is 25 times greater.

B. By analysing how many times greater and the factor that the dimensions are multiplied, you can notice the increase in area is factor². For example, when multiplied by a factor of 2, the new area is 4 times greater.

3 0

3 years ago

Answer the question for brainliest. If answered and explained well I’ll give brainliest

Step2247 [10]

Answer:

No

Step-by-step explanation:

n₁=4

n₂=9

n₃=16

We can see that all three numbers are perfect squares of one higher than the figure number

So, n₁₅=(15+1)²=16₂=256

256≠225

4 0

2 years ago

Kishore bought a basket containing 30 orangs for rs 310. If he sells the orangs at rs 12 each , what Would be it's profit or los

iris [78.8K]

Given : Kishore bought a basket containing 30 oranges for rs 310. If he sells the oranges at rs 12 each , what Would be it's profit or loss?

Solution :

Cost price of oranges = rs.310

As we know that each oranges cost rs.12

So, selling price of oranges is 30 × 12 = rs.360

Now, finding profit or loss

Profit = Selling Price - Cost Price

Profit = rs.360 - rs.310

Profit = rs.50

So, the profit is ₹50

7 0

2 years ago

Which pair of angles are vertical angles? ∠KRC and ∠CRH ∠MRC and ∠KRM ∠ERT and ∠MRC ∠KRE and ∠ERT Lines K H, M T, and E C inters

hram777 [196]

The best and the most correct answer among the choices provided by the question is the first choice. The pair of angles that are vertical is ∠KRC and ∠CRH ∠MRC and ∠KRM ∠ERT. I hope my answer has come to your help. God bless and have a nice day ahead!

8 0

3 years ago