ACT Math

Find the perimeter of an equilateral triangle whose side length is .

Explanation

To solve, simply multiply the side length by . Thus,

What is the value of m where:

$64^{\frac{n}{12}}=(m)4^{(\frac{1+m}{m+n})}$

-2

Explanation

If n=4, then 64(4/12)=64(1/3)=4. Then, 4=m4(1+m)/(m+4). If 2 is substituted for m, then 4=24(1+2)/(2+4)=241/2=2√4=22=4.

What is the volume of a sphere with a diameter of 6 in?

$36\pi\ in^{3}$

$216\pi\ in^{3}$

$108\pi\ in^{3}$

$72\pi\ in^{3}$

$288\pi\ in^{3}$

Explanation

The formula for the volume of a sphere is:

$V=\frac{4}{3}\pi r^{3}$

where = radius. The diameter is 6 in, so the radius will be 3 in.

What is the area of a triangle inscribed in a circle, in terms of the radius R of the circle?

Triangleinscribedincircle2

$3*\frac{\sqrt{3}}{4}*R^2$

$3*\sqrt{3}*R^2$

$5*\sqrt{2}*R^2$

$\sqrt{3}*R^2$

$4*\sqrt{3}*R^2$

Explanation

Draw 3 radii, and then 3 new lines that bisect the radii. You get six 30-60-90 triangles.

These triangles can be used ot find a side length $S = R*\sqrt{3}$ .

Using the formula for the area of an equilateral triangle in terms of its side, we get

$\frac{\sqrt{3}}{4}*S^2$

$\frac{\sqrt{3}}{4}(R*\sqrt3)^2 =$

$3*\frac{\sqrt{3}}{4} * R^2$

Simplify

Explanation

Simplifying an inequality like this is very simple. You merely need to treat it like an equation—just don't forget to keep the inequality sign.

First, subtract from both sides:

Then, divide by :

There is a line defined by the equation below:

There is a second line that passes through the point and is parallel to the line given above. What is the equation of this second line?

$y=-\frac{3}{4}x+2.75$

$y=-\frac{3}{4}x+1.25$

$y=\frac{3}{4}x+2.75$

$y=\frac{3}{4}x+1.25$

$y=\frac{3}{4}x+2.625$

Explanation

Parallel lines have the same slope. Solve for the slope in the first line by converting the equation to slope-intercept form.

3x + 4y = 12

4y = _–_3x + 12

y = –(3/4)x + 3

slope = _–_3/4

We know that the second line will also have a slope of _–_3/4, and we are given the point (1,2). We can set up an equation in slope-intercept form and use these values to solve for the y-intercept.

y = mx + b

2 = _–_3/4(1) + b

2 = _–_3/4 + b

b = 2 + 3/4 = 2.75

Plug the y-intercept back into the equation to get our final answer.

y = –(3/4)x + 2.75

Simplify

Explanation

Simplifying an inequality like this is very simple. You merely need to treat it like an equation—just don't forget to keep the inequality sign.

First, subtract from both sides:

Then, divide by :

A cube has a surface area of $150\textup{m}^2$ . What is its volume?

$125\textup{m}^3$

$1000\textup{m}^3$

$225\textup{m}^3$

$300\textup{m}^3$

Explanation

First, find the side lengths of the cube.

Recall that the surface area of the cube is given by the following equation:

$\text{Surface Area}=6s^2$ , where is the length of a side.

Plugging in the surface area given by the equation, we can then find the side length of the cube.

Now, recall that the volume of a cube is given by the following equation:

$\text{Volume}=s^3$

$\text{Volume}=5^3=125$

A store buys a new line of dresses for $15.00 each. They increase this price by 40% when they sell the dresses in the store. This week, the dresses are on sale for 20% off. What is the price of one dress this week?

$\dpi{100} \small $ 16.80$

$\dpi{100} \small $ 18.00$

$\dpi{100} \small $ 21.00$

$\dpi{100} \small $ 12.00$

$\dpi{100} \small $ 17.50$

Explanation

The price of one dress is now $16.80. The original price was $15.00. When the dresses were sold in the store, the price was increased by 40%, to $21.00

( $15+15\times 0.4=21$ )