(40 POINTS!)
What is the center of the circle that has a diameter whose endpoints are (-3, -5) and (-11, -7)

A. (-8, -1)
B. (-7, -6)
C. (-4, -6)

The author compare a melting ice cream cone to c. an atomic bomb.

In the essay ""How to Eat an Ice Cream Cone"" by Eugene Field, the author humorously compares the act of eating a melting ice cream cone to an atomic bomb.

The comparison is used to illustrate the chaotic and uncontrollable nature of trying to consume the ice cream before it melts, much like the destructive and rapid spread of an atomic bomb's explosion.

The author does not compare the melting ice cream cone to a runaway train, a machine gun, or a hand grenade, as these are not mentioned in the context of the essay.

The atomic bomb metaphor is used to emphasize the urgency and potential mess that can result from not eating the ice cream cone quickly enough, likening it to a catastrophic event that is difficult to contain once it starts."

To find the length of the pool, divide the volume (920 m³) by the product of the width (100 m) and depth (0.4 m), which gives a pool length of 23 meters.

The question asks to find the length of the pool given the volume, width, and depth of the pool. The volume of a rectangular prism (which is the shape of the pool) can be found using the formula Volume = length × width × depth. We can rearrange this formula to solve for the length by dividing the volume by the product of the width and depth.

Therefore, the length of the pool is calculated as follows:

So, the length of the pool is 23 meters.

Answer:

a)

i.the lower limit of the sum

: n=1

ii.the upper limit of the sum

: n=18

iii.the explicit formula of the sum

: [tex]a_n=7n-4[/tex]

b)  1,125 beads

Step-by-step explanation:

a) The number of beads per row of Dante's necklace is given by the series:

3 + 10 + 17 + 24 + ...

The first term of this series is [tex]a_1=3[/tex].

There is a constant difference of [tex]d=10-3=7[/tex].

To write this series in summation notation, we need to determine the explicit formula which is given by:

[tex]a_n=a_1+d(n-1)[/tex]

We plug in the values to get:

[tex]a_n=3+7(n-1)[/tex]

[tex]\implies a_n=3+7n-7[/tex]

[tex]\implies a_n=7n-4[/tex]

The summation notation is given by:

[tex]\sum^{18}_{n=1}(7n-4)[/tex]

b) The total number of beads in the necklace is given by the sum of the first 18 terms of the sequence.

This is given by [tex]S_n=\frac{n}{2}(2a_1+d(n-1))[/tex]

We substitute the values to obtain:

[tex]S_{18}=\frac{18}{2}(2\cdot3+7(18-1))[/tex]

[tex]S_{18}=9(6+119)[/tex]

[tex]S_{18}=9(125)[/tex]

[tex]S_{18}=1125[/tex]

Therefore the total number of beads is 1125

The arithmetic series 3, 10, 17, 24,... in which each term increases by 7 can be described in summation notation with a lower limit of 1, an upper limit of 18, and an explicit formula of 7n - 4. The total number of beads in Dante's 18-row necklace, computed using the arithmetic series sum formula, is 1125 beads.

The series described here is an arithmetic series, where each term increases by the same amount, 7 in this case.

a. To express this series in summation notation, you must know:

b. To find the total number of beads in the necklace, you use the formula for the sum of an arithmetic series. For the last term, l, we use the formula a+(n-1)d = 3+(18-1)*7 = 122. Hence, the total number of beads is S = 18/2 * (3 + 122) = 1125.

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Therefore option B is correct.

When a point is reflected across the line y = x, the x-coordinates and y-coordinates change their place. Similarly, when a point is reflected across the line y = -x, the x-coordinates and y-coordinates change their place and are negated. Therefore, The reflection of the point (x, y) across the line y = x is (y, x).

We have,

Coordinate of the vertex of the ΔFGH is F( -2 , -1 ) , G( 2 , 2 ) and H( 4 , -3 )

According to the question,

We know that when a point ( x , y ) reflected over y-axis , the y-coordinate remains the same but the x-coordinate is transformed into its opposite .i.e., ( -x , y )

So, Image of the Vertex F( -2 , -1 ) = F'( -(-2) , -1 ) = F'( 2 , -1 )

Image of the Vertex G( 2 , 2 ) = G'( -2 , 2 )

Image of the Vertex H( 4 , -3 ) = H'( -4 , -3 )

Hence, the  image of the vertexes is F'( 2 , -1 ) ,G'( -2 , 2 ) and H'( -4 , -3 )

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Answer:

c=7/3p

Step-by-step explanation:

Let's find the constant of proportionality.

in the proportional relationship between p, the number of hours parked, and c, the cost in dollars, one constant of proportionality is the cost per hour. It is the number we multiply by the number of hours to get the total cost.

p×?=c

px?=c

?= [tex]c/p[/tex]

= 7/3

the constant of proportionality is 7/3 this means we can multiply 7/3 by the numbers of hours to get the total cost

Now, let's write the equation: total cost

total cost=cost per hour×number of hours  

[tex]c=7/3p[/tex]

Final answer:

The ratio of the areas of two similar triangles with heights of 6 and 10 meters is 9/25.

Explanation:

To find the ratio of the areas of two similar triangles, △ABC and △DEF, with heights 6 meters and 10 meters respectively, we use the knowledge that the areas of similar triangles are proportional to the squares of the corresponding linear dimensions, such as their heights. For these triangles, the ratio of their heights is 6/10 or 3/5. Thus, the ratio of their areas is (3/5)2 or 9/25.

The answer is F) x+4y+7

Value of k = 10

Triangles are flat fields bounded by 3 intersecting sides and 3 angles

This side can be the same length or different.

There are two angles that can form:

From picture: ∠Y = Supplementary Angles:

115 ° + ∠Y = 180 °

∠Y = 180 ° - 115 °

∠Y = 65 °

As we know, the sum of angles in a triangle = 180 °

So from the picture, the sum of  ∠Y + ∠X + ∠Z = 180 °

65 ° + 6k + 10 ° + 4k + 5 ° = 180 ° (combine like terms)

65 ° + 10 ° + 5 ° + 6k + 4k = 180 °

80 ° + 10 k = 180 °

10 k = 180 ° -80 °

10 k = 100 °

k = 10

the Sin rule

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Answer:

The value of k is 10.

Step-by-step explanation:

Triangles are flat fields bounded by 3 intersecting sides and 3 angles

This side can be the same length or different.

There are two angles that can form:

Supplementary Angles: if both angles are added = 180 °

Complementary Angles: if both angles are added = 90 °

From picture: ∠Y = Supplementary Angles:

115 ° + ∠Y = 180 °

∠Y = 180 ° - 115 °

∠Y = 65 °

As we know, the sum of all interior angles of a triangle = 180 °

So from the picture, the sum of  ∠Y + ∠X + ∠Z = 180 °

65 ° + 6k + 10 ° + 4k + 5 ° = 180 ° (combine like terms)

65 ° + 10 ° + 5 ° + 6k + 4k = 180 °

80 ° + 10 k = 180 °

10 k = 180 ° -80 °

10 k = 100 °

k = 10

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Answer:

The center of circle N is (0, 10).

Circle M and circle N are similar.

Step-by-step explanation:

It is given circle N is translation of circle M 1 unit left .The radius of circle M is (1,10) when it is translated 1 units left the circle N will have the center (0,10).The y value will remain same while the x coordinated is reduced by 1 that is 1-1=0.The center of circle N is (0, 10)

As the circle N is a dilation of circle M by scale factor 3 so the radius of the two circle will not be same and hence the diameter will also be different.

In translation  we have a similar figure ,So circle M and N will be similar.

Final answer:

If two angles of one triangle are congruent to two angles in another triangle, the third angles must also be congruent since the sum of angles in any triangle must equal 180 degrees. This principle is supported by geometric theorems about the properties of triangles and the requirements for their congruence and similarity.

Explanation:

If two angles of one triangle are congruent to two angles in another triangle, then the third angles in both triangles must also be congruent. This is because the sum of the angles in any triangle must equal 180 degrees or two right angles. Since two angles are congruent in both triangles, whatever is left from the 180 degrees after subtracting these two angles must be what is left for the third angle, making them congruent by necessity.

There are several related theorems in geometry that confirm this. Theorem 11, also known as the Side-Angle-Side (SAS) Postulate, states that if one side and the two adjacent angles of two triangles are congruent, then the triangles are congruent. While this theorem refers mainly to the congruency of triangles, it supports our understanding that if two angles are congruent, the third must be as well to satisfy the congruence of all three angles.

Furthermore, in the context of similar triangles, when two triangles have congruent angles, they are similar, meaning all their angles are congruent, and their sides are in proportion. However, if we only consider the angles, as in the original question, knowing that two angles are congruent across two triangles is enough to deduct that the third angles will be congruent, regardless of side lengths or overall similarity.

Answer:

Step-by-step explanation:

Since we have been given two points, P(4,1,1) and Q(0,-7,9), we can take the vector between the two toget the directional vector.

So, vector PQ = v = <-4, -8, 8>. Now, we find its UNIT VECTOR.

Since the UNIT VECTOR is just the vector divided by its magnitude, we get that the unit vector u = v / |v|

|v| = [tex]\sqrt{(-4)^2+(-8)^2+(8^2)}[/tex]  = [tex]\sqrt{144}[/tex] = 12

Dividing each part of the original vector v by 12 gives us the unit vector.

Now, we take the partial derivatives, Fx, Fy, and Fz.

Fx = [tex]y^2z^3[/tex]  

Fy = [tex]x(2y)z^3[/tex]

Fz = [tex]xy^2(3z^2)[/tex]

plugging in the original point P(4,1,1) gives us the following values for the gradient, which is just the vector of the partial derivatives.

ΔF = <1,8,12>

Now we just use the equation Duf = ΔF * u, where we take the DOT PRODUCT of the gradient at the given point with the direction unit vector.

This gives us <1,8,12> * <-1/3, -2/3, 2/3> = 7/3

The directional derivative of [tex]\( f \)[/tex] at point [tex]\( P(4, 1, 1) \)[/tex] in the direction of [tex]\( Q(0, -7, 9) \)[/tex] is [tex]\( \frac{7}{\sqrt{6}} \).[/tex]

To find the directional derivative of the function [tex]\(f(x, y, z) = xy^2z^3\)[/tex] at the point [tex]\(P(4, 1, 1)\)[/tex] in the direction of [tex]\(Q(0, -7, 9)\)[/tex], we'll use the formula for directional derivative:

[tex]\[ D_{\mathbf{u}} f(x, y, z) = \nabla f \cdot \mathbf{u} \][/tex]

where [tex]\( \nabla f \)[/tex] is the gradient vector of [tex]\( f \)[/tex] and [tex]\( \mathbf{u} \)[/tex] is the unit vector in the direction of [tex]\( Q \).[/tex]

First, let's find the gradient vector [tex]\( \nabla f \):[/tex]

[tex]\[ \nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \][/tex]

[tex]\[ = \left( y^2z^3, 2xyz^3, 3xy^2z^2 \right) \][/tex]

Now, evaluate [tex]\( \nabla f \)[/tex] at the point [tex]\( P(4, 1, 1) \):[/tex]

[tex]\[ \nabla f(P) = \left( (1)^2(1)^3, 2(4)(1)(1)^3, 3(4)(1)^2(1)^2 \right) \][/tex]

[tex]\[ = (1, 8, 12) \][/tex]

Next, find the unit vector in the direction of [tex]\( Q \):[/tex]

[tex]\[ \mathbf{u} = \frac{\mathbf{Q} - \mathbf{P}}{\|\mathbf{Q} - \mathbf{P}\|} \][/tex]

[tex]\[ = \frac{(0 - 4, -7 - 1, 9 - 1)}{\sqrt{(0 - 4)^2 + (-7 - 1)^2 + (9 - 1)^2}} \][/tex]

[tex]\[ = \frac{(-4, -8, 8)}{\sqrt{16 + 64 + 64}} \][/tex]

[tex]\[ = \frac{(-4, -8, 8)}{4\sqrt{6}} \][/tex]

[tex]\[ = \left( -\frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}}, \frac{2}{\sqrt{6}} \right) \][/tex]

Now, calculate the dot product [tex]\( \nabla f \cdot \mathbf{u} \):[/tex]

[tex]\[ D_{\mathbf{u}} f(P) = \nabla f(P) \cdot \mathbf{u} \][/tex]

[tex]\[ = (1, 8, 12) \cdot \left( -\frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}}, \frac{2}{\sqrt{6}} \right) \][/tex]

[tex]\[ = -\frac{1}{\sqrt{6}} + \left(-\frac{16}{\sqrt{6}}\right) + \frac{24}{\sqrt{6}} \][/tex]

[tex]\[ = \frac{7}{\sqrt{6}} \][/tex]

Therefore, the directional derivative of [tex]\( f \)[/tex] at point [tex]\( P(4, 1, 1) \)[/tex] in the direction of [tex]\( Q(0, -7, 9) \)[/tex] is [tex]\( \frac{7}{\sqrt{6}} \).[/tex]

Answer:

1,-4

0,4

3,-724

Step-by-step explanation: