Question

The base of a triangle is four less than three times the height. The area of the triangle is 38 square centimeters. A = (b×h)/2

Answer 1

An expression for the base of the triangle: Response area

Equation for the area of the triangle, substituting in for the base: Response area

The first step in solving for h: Response area

Answer 2

show

Answer 3

To find the base and height of the triangle, we can start by using the formula for the area of a triangle:

\[
A = \frac{b \times h}{2}
\]

We know the area \( A = 38 \) cm². Thus, we can write:

\[
38 = \frac{b \times h}{2}
\]

Multiplying both sides by 2 to eliminate the fraction gives us:

\[
76 = b \times h
\]

Next, we also know the relationship between the base \( b \) and the height \( h \). The problem states that the base is four less than three times the height:

\[
b = 3h - 4
\]

Now we can substitute this expression for \( b \) into the area equation:

\[
76 = (3h - 4) \times h
\]

Expanding this gives:

\[
76 = 3h^2 - 4h
\]

Rearranging the equation gives us a standard quadratic form:

\[
3h^2 - 4h - 76 = 0
\]

Next, we can use the quadratic formula \( h = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Here, \( a = 3 \), \( b = -4 \), and \( c = -76 \).

Calculating the discriminant:

\[
b^2 - 4ac = (-4)^2 - 4 \times 3 \times (-76) = 16 + 912 = 928
\]

Now we can apply the quadratic formula:

\[
h = \frac{-(-4) \pm \sqrt{928}}{2 \times 3} = \frac{4 \pm \sqrt{928}}{6}
\]

Calculating \( \sqrt{928} \):

\[
\sqrt{928} \approx 30.5
\]

Now we substitute this back into the equation for \( h \):

\[
h = \frac{4 \pm 30.5}{6}
\]

This gives us two potential solutions:

1. \( h = \frac{34.5}{6} \approx 5.75 \)
2. \( h = \frac{-26.5}{6} \approx -4.42 \) (not a valid solution, since height cannot be negative)

Thus, we take the valid solution:

\[
h \approx 5.75
\]

Now we can find \( b \):

\[
b = 3h - 4 = 3(5.75) - 4 = 17.25 - 4 = 13.25
\]

So the height of the triangle is approximately \( 5.75 \) cm and the base is approximately \( 13.25 \) cm.

To summarize:

- Height \( h \approx 5.75 \) cm
- Base \( b \approx 13.25 \) cm

Answer 4

Sure! Let's break this down step by step.

### Expression for the Base of the Triangle
The base \( b \) of the triangle can be expressed in terms of the height \( h \) as given in the problem:

\[
b = 3h - 4
\]

### Equation for the Area of the Triangle, Substituting in for the Base
Using the formula for the area of a triangle \( A = \frac{b \times h}{2} \) and substituting the expression for the base \( b \):

\[
38 = \frac{(3h - 4) \times h}{2}
\]

### The First Step in Solving for \( h \)
To eliminate the fraction, multiply both sides of the equation by 2:

\[
76 = (3h - 4) \times h
\]

This is the first step in solving for \( h \).

Answer 5

Certainly! Let's go through the steps together, showing each part of the process clearly.

### Step 1: Expression for the Base of the Triangle
Given that the base \( b \) of the triangle is four less than three times the height \( h \), we can express this as:

\[
b = 3h - 4
\]

### Step 2: Equation for the Area of the Triangle, Substituting in for the Base
The formula for the area \( A \) of a triangle is given by:

\[
A = \frac{b \times h}{2}
\]

We know that the area \( A \) is 38 square centimeters. Substituting the expression for \( b \) into the area formula:

\[
38 = \frac{(3h - 4) \times h}{2}
\]

### Step 3: Eliminate the Fraction
The next step is to eliminate the fraction by multiplying both sides of the equation by 2:

\[
2 \times 38 = (3h - 4) \times h
\]

Calculating the left side gives:

\[
76 = (3h - 4) \times h
\]

This is the equation we'll solve for \( h \).

### Step 4: Expand and Rearrange
Now we can expand and rearrange the equation:

\[
76 = 3h^2 - 4h
\]

Rearranging it into standard quadratic form provides us with:

\[
3h^2 - 4h - 76 = 0
\]

### Summary of Steps
1. **Expression for the Base**: \( b = 3h - 4 \)
2. **Area Equation Substituted**: \( 38 = \frac{(3h - 4) \times h}{2} \)
3. **First Step in Solving for \( h \)**: \( 76 = (3h - 4) \times h \)

From here, you can proceed to use the quadratic formula or other methods to solve for \( h \). Would you like to continue working through the solution?